620 lines
24 KiB
JavaScript
620 lines
24 KiB
JavaScript
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import { isConstantNode, typeOf } from '../../utils/is.js';
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import { factory } from '../../utils/factory.js';
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var name = 'derivative';
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var dependencies = ['typed', 'config', 'parse', 'simplify', 'equal', 'isZero', 'numeric', 'ConstantNode', 'FunctionNode', 'OperatorNode', 'ParenthesisNode', 'SymbolNode'];
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export var createDerivative = /* #__PURE__ */factory(name, dependencies, _ref => {
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var {
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typed,
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config,
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parse,
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simplify,
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equal,
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isZero,
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numeric,
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ConstantNode,
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FunctionNode,
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OperatorNode,
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ParenthesisNode,
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SymbolNode
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} = _ref;
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/**
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* Takes the derivative of an expression expressed in parser Nodes.
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* The derivative will be taken over the supplied variable in the
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* second parameter. If there are multiple variables in the expression,
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* it will return a partial derivative.
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*
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* This uses rules of differentiation which can be found here:
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*
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* - [Differentiation rules (Wikipedia)](https://en.wikipedia.org/wiki/Differentiation_rules)
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*
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* Syntax:
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*
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* derivative(expr, variable)
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* derivative(expr, variable, options)
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*
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* Examples:
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*
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* math.derivative('x^2', 'x') // Node {2 * x}
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* math.derivative('x^2', 'x', {simplify: false}) // Node {2 * 1 * x ^ (2 - 1)
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* math.derivative('sin(2x)', 'x')) // Node {2 * cos(2 * x)}
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* math.derivative('2*x', 'x').evaluate() // number 2
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* math.derivative('x^2', 'x').evaluate({x: 4}) // number 8
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* const f = math.parse('x^2')
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* const x = math.parse('x')
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* math.derivative(f, x) // Node {2 * x}
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*
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* See also:
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*
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* simplify, parse, evaluate
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*
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* @param {Node | string} expr The expression to differentiate
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* @param {SymbolNode | string} variable The variable over which to differentiate
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* @param {{simplify: boolean}} [options]
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* There is one option available, `simplify`, which
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* is true by default. When false, output will not
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* be simplified.
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* @return {ConstantNode | SymbolNode | ParenthesisNode | FunctionNode | OperatorNode} The derivative of `expr`
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*/
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var derivative = typed('derivative', {
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'Node, SymbolNode, Object': function NodeSymbolNodeObject(expr, variable, options) {
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var constNodes = {};
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constTag(constNodes, expr, variable.name);
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var res = _derivative(expr, constNodes);
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return options.simplify ? simplify(res) : res;
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},
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'Node, SymbolNode': function NodeSymbolNode(expr, variable) {
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return this(expr, variable, {
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simplify: true
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});
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},
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'string, SymbolNode': function stringSymbolNode(expr, variable) {
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return this(parse(expr), variable);
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},
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'string, SymbolNode, Object': function stringSymbolNodeObject(expr, variable, options) {
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return this(parse(expr), variable, options);
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},
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'string, string': function stringString(expr, variable) {
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return this(parse(expr), parse(variable));
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},
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'string, string, Object': function stringStringObject(expr, variable, options) {
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return this(parse(expr), parse(variable), options);
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},
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'Node, string': function NodeString(expr, variable) {
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return this(expr, parse(variable));
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},
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'Node, string, Object': function NodeStringObject(expr, variable, options) {
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return this(expr, parse(variable), options);
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} // TODO: replace the 8 signatures above with 4 as soon as typed-function supports optional arguments
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/* TODO: implement and test syntax with order of derivatives -> implement as an option {order: number}
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'Node, SymbolNode, ConstantNode': function (expr, variable, {order}) {
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let res = expr
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for (let i = 0; i < order; i++) {
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let constNodes = {}
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constTag(constNodes, expr, variable.name)
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res = _derivative(res, constNodes)
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}
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return res
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}
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*/
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});
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derivative._simplify = true;
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derivative.toTex = function (deriv) {
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return _derivTex.apply(null, deriv.args);
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}; // FIXME: move the toTex method of derivative to latex.js. Difficulty is that it relies on parse.
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// NOTE: the optional "order" parameter here is currently unused
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var _derivTex = typed('_derivTex', {
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'Node, SymbolNode': function NodeSymbolNode(expr, x) {
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if (isConstantNode(expr) && typeOf(expr.value) === 'string') {
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return _derivTex(parse(expr.value).toString(), x.toString(), 1);
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} else {
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return _derivTex(expr.toString(), x.toString(), 1);
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}
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},
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'Node, ConstantNode': function NodeConstantNode(expr, x) {
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if (typeOf(x.value) === 'string') {
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return _derivTex(expr, parse(x.value));
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} else {
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throw new Error("The second parameter to 'derivative' is a non-string constant");
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}
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},
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'Node, SymbolNode, ConstantNode': function NodeSymbolNodeConstantNode(expr, x, order) {
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return _derivTex(expr.toString(), x.name, order.value);
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},
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'string, string, number': function stringStringNumber(expr, x, order) {
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var d;
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if (order === 1) {
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d = '{d\\over d' + x + '}';
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} else {
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d = '{d^{' + order + '}\\over d' + x + '^{' + order + '}}';
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}
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return d + "\\left[".concat(expr, "\\right]");
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}
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});
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/**
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* Does a depth-first search on the expression tree to identify what Nodes
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* are constants (e.g. 2 + 2), and stores the ones that are constants in
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* constNodes. Classification is done as follows:
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*
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* 1. ConstantNodes are constants.
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* 2. If there exists a SymbolNode, of which we are differentiating over,
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* in the subtree it is not constant.
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*
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* @param {Object} constNodes Holds the nodes that are constant
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* @param {ConstantNode | SymbolNode | ParenthesisNode | FunctionNode | OperatorNode} node
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* @param {string} varName Variable that we are differentiating
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* @return {boolean} if node is constant
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*/
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// TODO: can we rewrite constTag into a pure function?
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var constTag = typed('constTag', {
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'Object, ConstantNode, string': function ObjectConstantNodeString(constNodes, node) {
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constNodes[node] = true;
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return true;
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},
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'Object, SymbolNode, string': function ObjectSymbolNodeString(constNodes, node, varName) {
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// Treat other variables like constants. For reasoning, see:
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// https://en.wikipedia.org/wiki/Partial_derivative
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if (node.name !== varName) {
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constNodes[node] = true;
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return true;
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}
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return false;
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},
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'Object, ParenthesisNode, string': function ObjectParenthesisNodeString(constNodes, node, varName) {
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return constTag(constNodes, node.content, varName);
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},
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'Object, FunctionAssignmentNode, string': function ObjectFunctionAssignmentNodeString(constNodes, node, varName) {
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if (node.params.indexOf(varName) === -1) {
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constNodes[node] = true;
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return true;
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}
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return constTag(constNodes, node.expr, varName);
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},
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'Object, FunctionNode | OperatorNode, string': function ObjectFunctionNodeOperatorNodeString(constNodes, node, varName) {
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if (node.args.length > 0) {
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var isConst = constTag(constNodes, node.args[0], varName);
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for (var i = 1; i < node.args.length; ++i) {
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isConst = constTag(constNodes, node.args[i], varName) && isConst;
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}
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if (isConst) {
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constNodes[node] = true;
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return true;
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}
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}
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return false;
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}
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});
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/**
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* Applies differentiation rules.
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*
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* @param {ConstantNode | SymbolNode | ParenthesisNode | FunctionNode | OperatorNode} node
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* @param {Object} constNodes Holds the nodes that are constant
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* @return {ConstantNode | SymbolNode | ParenthesisNode | FunctionNode | OperatorNode} The derivative of `expr`
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*/
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var _derivative = typed('_derivative', {
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'ConstantNode, Object': function ConstantNodeObject(node) {
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return createConstantNode(0);
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},
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'SymbolNode, Object': function SymbolNodeObject(node, constNodes) {
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if (constNodes[node] !== undefined) {
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return createConstantNode(0);
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}
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return createConstantNode(1);
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},
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'ParenthesisNode, Object': function ParenthesisNodeObject(node, constNodes) {
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return new ParenthesisNode(_derivative(node.content, constNodes));
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},
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'FunctionAssignmentNode, Object': function FunctionAssignmentNodeObject(node, constNodes) {
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if (constNodes[node] !== undefined) {
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return createConstantNode(0);
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}
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return _derivative(node.expr, constNodes);
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},
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'FunctionNode, Object': function FunctionNodeObject(node, constNodes) {
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if (node.args.length !== 1) {
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funcArgsCheck(node);
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}
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if (constNodes[node] !== undefined) {
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return createConstantNode(0);
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}
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var arg0 = node.args[0];
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var arg1;
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var div = false; // is output a fraction?
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var negative = false; // is output negative?
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var funcDerivative;
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switch (node.name) {
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case 'cbrt':
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// d/dx(cbrt(x)) = 1 / (3x^(2/3))
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div = true;
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funcDerivative = new OperatorNode('*', 'multiply', [createConstantNode(3), new OperatorNode('^', 'pow', [arg0, new OperatorNode('/', 'divide', [createConstantNode(2), createConstantNode(3)])])]);
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break;
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case 'sqrt':
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case 'nthRoot':
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// d/dx(sqrt(x)) = 1 / (2*sqrt(x))
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if (node.args.length === 1) {
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div = true;
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funcDerivative = new OperatorNode('*', 'multiply', [createConstantNode(2), new FunctionNode('sqrt', [arg0])]);
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} else if (node.args.length === 2) {
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// Rearrange from nthRoot(x, a) -> x^(1/a)
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arg1 = new OperatorNode('/', 'divide', [createConstantNode(1), node.args[1]]); // Is a variable?
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constNodes[arg1] = constNodes[node.args[1]];
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return _derivative(new OperatorNode('^', 'pow', [arg0, arg1]), constNodes);
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}
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break;
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case 'log10':
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arg1 = createConstantNode(10);
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/* fall through! */
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case 'log':
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if (!arg1 && node.args.length === 1) {
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// d/dx(log(x)) = 1 / x
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funcDerivative = arg0.clone();
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div = true;
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} else if (node.args.length === 1 && arg1 || node.args.length === 2 && constNodes[node.args[1]] !== undefined) {
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// d/dx(log(x, c)) = 1 / (x*ln(c))
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funcDerivative = new OperatorNode('*', 'multiply', [arg0.clone(), new FunctionNode('log', [arg1 || node.args[1]])]);
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div = true;
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} else if (node.args.length === 2) {
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// d/dx(log(f(x), g(x))) = d/dx(log(f(x)) / log(g(x)))
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return _derivative(new OperatorNode('/', 'divide', [new FunctionNode('log', [arg0]), new FunctionNode('log', [node.args[1]])]), constNodes);
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}
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break;
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case 'pow':
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constNodes[arg1] = constNodes[node.args[1]]; // Pass to pow operator node parser
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return _derivative(new OperatorNode('^', 'pow', [arg0, node.args[1]]), constNodes);
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case 'exp':
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// d/dx(e^x) = e^x
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funcDerivative = new FunctionNode('exp', [arg0.clone()]);
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break;
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case 'sin':
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// d/dx(sin(x)) = cos(x)
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funcDerivative = new FunctionNode('cos', [arg0.clone()]);
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break;
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case 'cos':
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// d/dx(cos(x)) = -sin(x)
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funcDerivative = new OperatorNode('-', 'unaryMinus', [new FunctionNode('sin', [arg0.clone()])]);
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break;
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case 'tan':
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// d/dx(tan(x)) = sec(x)^2
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funcDerivative = new OperatorNode('^', 'pow', [new FunctionNode('sec', [arg0.clone()]), createConstantNode(2)]);
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break;
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case 'sec':
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// d/dx(sec(x)) = sec(x)tan(x)
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funcDerivative = new OperatorNode('*', 'multiply', [node, new FunctionNode('tan', [arg0.clone()])]);
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break;
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case 'csc':
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// d/dx(csc(x)) = -csc(x)cot(x)
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negative = true;
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funcDerivative = new OperatorNode('*', 'multiply', [node, new FunctionNode('cot', [arg0.clone()])]);
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break;
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case 'cot':
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// d/dx(cot(x)) = -csc(x)^2
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negative = true;
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funcDerivative = new OperatorNode('^', 'pow', [new FunctionNode('csc', [arg0.clone()]), createConstantNode(2)]);
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break;
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case 'asin':
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// d/dx(asin(x)) = 1 / sqrt(1 - x^2)
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div = true;
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funcDerivative = new FunctionNode('sqrt', [new OperatorNode('-', 'subtract', [createConstantNode(1), new OperatorNode('^', 'pow', [arg0.clone(), createConstantNode(2)])])]);
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break;
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case 'acos':
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// d/dx(acos(x)) = -1 / sqrt(1 - x^2)
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div = true;
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negative = true;
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funcDerivative = new FunctionNode('sqrt', [new OperatorNode('-', 'subtract', [createConstantNode(1), new OperatorNode('^', 'pow', [arg0.clone(), createConstantNode(2)])])]);
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break;
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case 'atan':
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// d/dx(atan(x)) = 1 / (x^2 + 1)
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div = true;
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funcDerivative = new OperatorNode('+', 'add', [new OperatorNode('^', 'pow', [arg0.clone(), createConstantNode(2)]), createConstantNode(1)]);
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break;
|
||
|
|
|
||
|
|
case 'asec':
|
||
|
|
// d/dx(asec(x)) = 1 / (|x|*sqrt(x^2 - 1))
|
||
|
|
div = true;
|
||
|
|
funcDerivative = new OperatorNode('*', 'multiply', [new FunctionNode('abs', [arg0.clone()]), new FunctionNode('sqrt', [new OperatorNode('-', 'subtract', [new OperatorNode('^', 'pow', [arg0.clone(), createConstantNode(2)]), createConstantNode(1)])])]);
|
||
|
|
break;
|
||
|
|
|
||
|
|
case 'acsc':
|
||
|
|
// d/dx(acsc(x)) = -1 / (|x|*sqrt(x^2 - 1))
|
||
|
|
div = true;
|
||
|
|
negative = true;
|
||
|
|
funcDerivative = new OperatorNode('*', 'multiply', [new FunctionNode('abs', [arg0.clone()]), new FunctionNode('sqrt', [new OperatorNode('-', 'subtract', [new OperatorNode('^', 'pow', [arg0.clone(), createConstantNode(2)]), createConstantNode(1)])])]);
|
||
|
|
break;
|
||
|
|
|
||
|
|
case 'acot':
|
||
|
|
// d/dx(acot(x)) = -1 / (x^2 + 1)
|
||
|
|
div = true;
|
||
|
|
negative = true;
|
||
|
|
funcDerivative = new OperatorNode('+', 'add', [new OperatorNode('^', 'pow', [arg0.clone(), createConstantNode(2)]), createConstantNode(1)]);
|
||
|
|
break;
|
||
|
|
|
||
|
|
case 'sinh':
|
||
|
|
// d/dx(sinh(x)) = cosh(x)
|
||
|
|
funcDerivative = new FunctionNode('cosh', [arg0.clone()]);
|
||
|
|
break;
|
||
|
|
|
||
|
|
case 'cosh':
|
||
|
|
// d/dx(cosh(x)) = sinh(x)
|
||
|
|
funcDerivative = new FunctionNode('sinh', [arg0.clone()]);
|
||
|
|
break;
|
||
|
|
|
||
|
|
case 'tanh':
|
||
|
|
// d/dx(tanh(x)) = sech(x)^2
|
||
|
|
funcDerivative = new OperatorNode('^', 'pow', [new FunctionNode('sech', [arg0.clone()]), createConstantNode(2)]);
|
||
|
|
break;
|
||
|
|
|
||
|
|
case 'sech':
|
||
|
|
// d/dx(sech(x)) = -sech(x)tanh(x)
|
||
|
|
negative = true;
|
||
|
|
funcDerivative = new OperatorNode('*', 'multiply', [node, new FunctionNode('tanh', [arg0.clone()])]);
|
||
|
|
break;
|
||
|
|
|
||
|
|
case 'csch':
|
||
|
|
// d/dx(csch(x)) = -csch(x)coth(x)
|
||
|
|
negative = true;
|
||
|
|
funcDerivative = new OperatorNode('*', 'multiply', [node, new FunctionNode('coth', [arg0.clone()])]);
|
||
|
|
break;
|
||
|
|
|
||
|
|
case 'coth':
|
||
|
|
// d/dx(coth(x)) = -csch(x)^2
|
||
|
|
negative = true;
|
||
|
|
funcDerivative = new OperatorNode('^', 'pow', [new FunctionNode('csch', [arg0.clone()]), createConstantNode(2)]);
|
||
|
|
break;
|
||
|
|
|
||
|
|
case 'asinh':
|
||
|
|
// d/dx(asinh(x)) = 1 / sqrt(x^2 + 1)
|
||
|
|
div = true;
|
||
|
|
funcDerivative = new FunctionNode('sqrt', [new OperatorNode('+', 'add', [new OperatorNode('^', 'pow', [arg0.clone(), createConstantNode(2)]), createConstantNode(1)])]);
|
||
|
|
break;
|
||
|
|
|
||
|
|
case 'acosh':
|
||
|
|
// d/dx(acosh(x)) = 1 / sqrt(x^2 - 1); XXX potentially only for x >= 1 (the real spectrum)
|
||
|
|
div = true;
|
||
|
|
funcDerivative = new FunctionNode('sqrt', [new OperatorNode('-', 'subtract', [new OperatorNode('^', 'pow', [arg0.clone(), createConstantNode(2)]), createConstantNode(1)])]);
|
||
|
|
break;
|
||
|
|
|
||
|
|
case 'atanh':
|
||
|
|
// d/dx(atanh(x)) = 1 / (1 - x^2)
|
||
|
|
div = true;
|
||
|
|
funcDerivative = new OperatorNode('-', 'subtract', [createConstantNode(1), new OperatorNode('^', 'pow', [arg0.clone(), createConstantNode(2)])]);
|
||
|
|
break;
|
||
|
|
|
||
|
|
case 'asech':
|
||
|
|
// d/dx(asech(x)) = -1 / (x*sqrt(1 - x^2))
|
||
|
|
div = true;
|
||
|
|
negative = true;
|
||
|
|
funcDerivative = new OperatorNode('*', 'multiply', [arg0.clone(), new FunctionNode('sqrt', [new OperatorNode('-', 'subtract', [createConstantNode(1), new OperatorNode('^', 'pow', [arg0.clone(), createConstantNode(2)])])])]);
|
||
|
|
break;
|
||
|
|
|
||
|
|
case 'acsch':
|
||
|
|
// d/dx(acsch(x)) = -1 / (|x|*sqrt(x^2 + 1))
|
||
|
|
div = true;
|
||
|
|
negative = true;
|
||
|
|
funcDerivative = new OperatorNode('*', 'multiply', [new FunctionNode('abs', [arg0.clone()]), new FunctionNode('sqrt', [new OperatorNode('+', 'add', [new OperatorNode('^', 'pow', [arg0.clone(), createConstantNode(2)]), createConstantNode(1)])])]);
|
||
|
|
break;
|
||
|
|
|
||
|
|
case 'acoth':
|
||
|
|
// d/dx(acoth(x)) = -1 / (1 - x^2)
|
||
|
|
div = true;
|
||
|
|
negative = true;
|
||
|
|
funcDerivative = new OperatorNode('-', 'subtract', [createConstantNode(1), new OperatorNode('^', 'pow', [arg0.clone(), createConstantNode(2)])]);
|
||
|
|
break;
|
||
|
|
|
||
|
|
case 'abs':
|
||
|
|
// d/dx(abs(x)) = abs(x)/x
|
||
|
|
funcDerivative = new OperatorNode('/', 'divide', [new FunctionNode(new SymbolNode('abs'), [arg0.clone()]), arg0.clone()]);
|
||
|
|
break;
|
||
|
|
|
||
|
|
case 'gamma': // Needs digamma function, d/dx(gamma(x)) = gamma(x)digamma(x)
|
||
|
|
|
||
|
|
default:
|
||
|
|
throw new Error('Function "' + node.name + '" is not supported by derivative, or a wrong number of arguments is passed');
|
||
|
|
}
|
||
|
|
|
||
|
|
var op, func;
|
||
|
|
|
||
|
|
if (div) {
|
||
|
|
op = '/';
|
||
|
|
func = 'divide';
|
||
|
|
} else {
|
||
|
|
op = '*';
|
||
|
|
func = 'multiply';
|
||
|
|
}
|
||
|
|
/* Apply chain rule to all functions:
|
||
|
|
F(x) = f(g(x))
|
||
|
|
F'(x) = g'(x)*f'(g(x)) */
|
||
|
|
|
||
|
|
|
||
|
|
var chainDerivative = _derivative(arg0, constNodes);
|
||
|
|
|
||
|
|
if (negative) {
|
||
|
|
chainDerivative = new OperatorNode('-', 'unaryMinus', [chainDerivative]);
|
||
|
|
}
|
||
|
|
|
||
|
|
return new OperatorNode(op, func, [chainDerivative, funcDerivative]);
|
||
|
|
},
|
||
|
|
'OperatorNode, Object': function OperatorNodeObject(node, constNodes) {
|
||
|
|
if (constNodes[node] !== undefined) {
|
||
|
|
return createConstantNode(0);
|
||
|
|
}
|
||
|
|
|
||
|
|
if (node.op === '+') {
|
||
|
|
// d/dx(sum(f(x)) = sum(f'(x))
|
||
|
|
return new OperatorNode(node.op, node.fn, node.args.map(function (arg) {
|
||
|
|
return _derivative(arg, constNodes);
|
||
|
|
}));
|
||
|
|
}
|
||
|
|
|
||
|
|
if (node.op === '-') {
|
||
|
|
// d/dx(+/-f(x)) = +/-f'(x)
|
||
|
|
if (node.isUnary()) {
|
||
|
|
return new OperatorNode(node.op, node.fn, [_derivative(node.args[0], constNodes)]);
|
||
|
|
} // Linearity of differentiation, d/dx(f(x) +/- g(x)) = f'(x) +/- g'(x)
|
||
|
|
|
||
|
|
|
||
|
|
if (node.isBinary()) {
|
||
|
|
return new OperatorNode(node.op, node.fn, [_derivative(node.args[0], constNodes), _derivative(node.args[1], constNodes)]);
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
if (node.op === '*') {
|
||
|
|
// d/dx(c*f(x)) = c*f'(x)
|
||
|
|
var constantTerms = node.args.filter(function (arg) {
|
||
|
|
return constNodes[arg] !== undefined;
|
||
|
|
});
|
||
|
|
|
||
|
|
if (constantTerms.length > 0) {
|
||
|
|
var nonConstantTerms = node.args.filter(function (arg) {
|
||
|
|
return constNodes[arg] === undefined;
|
||
|
|
});
|
||
|
|
var nonConstantNode = nonConstantTerms.length === 1 ? nonConstantTerms[0] : new OperatorNode('*', 'multiply', nonConstantTerms);
|
||
|
|
var newArgs = constantTerms.concat(_derivative(nonConstantNode, constNodes));
|
||
|
|
return new OperatorNode('*', 'multiply', newArgs);
|
||
|
|
} // Product Rule, d/dx(f(x)*g(x)) = f'(x)*g(x) + f(x)*g'(x)
|
||
|
|
|
||
|
|
|
||
|
|
return new OperatorNode('+', 'add', node.args.map(function (argOuter) {
|
||
|
|
return new OperatorNode('*', 'multiply', node.args.map(function (argInner) {
|
||
|
|
return argInner === argOuter ? _derivative(argInner, constNodes) : argInner.clone();
|
||
|
|
}));
|
||
|
|
}));
|
||
|
|
}
|
||
|
|
|
||
|
|
if (node.op === '/' && node.isBinary()) {
|
||
|
|
var arg0 = node.args[0];
|
||
|
|
var arg1 = node.args[1]; // d/dx(f(x) / c) = f'(x) / c
|
||
|
|
|
||
|
|
if (constNodes[arg1] !== undefined) {
|
||
|
|
return new OperatorNode('/', 'divide', [_derivative(arg0, constNodes), arg1]);
|
||
|
|
} // Reciprocal Rule, d/dx(c / f(x)) = -c(f'(x)/f(x)^2)
|
||
|
|
|
||
|
|
|
||
|
|
if (constNodes[arg0] !== undefined) {
|
||
|
|
return new OperatorNode('*', 'multiply', [new OperatorNode('-', 'unaryMinus', [arg0]), new OperatorNode('/', 'divide', [_derivative(arg1, constNodes), new OperatorNode('^', 'pow', [arg1.clone(), createConstantNode(2)])])]);
|
||
|
|
} // Quotient rule, d/dx(f(x) / g(x)) = (f'(x)g(x) - f(x)g'(x)) / g(x)^2
|
||
|
|
|
||
|
|
|
||
|
|
return new OperatorNode('/', 'divide', [new OperatorNode('-', 'subtract', [new OperatorNode('*', 'multiply', [_derivative(arg0, constNodes), arg1.clone()]), new OperatorNode('*', 'multiply', [arg0.clone(), _derivative(arg1, constNodes)])]), new OperatorNode('^', 'pow', [arg1.clone(), createConstantNode(2)])]);
|
||
|
|
}
|
||
|
|
|
||
|
|
if (node.op === '^' && node.isBinary()) {
|
||
|
|
var _arg = node.args[0];
|
||
|
|
var _arg2 = node.args[1];
|
||
|
|
|
||
|
|
if (constNodes[_arg] !== undefined) {
|
||
|
|
// If is secretly constant; 0^f(x) = 1 (in JS), 1^f(x) = 1
|
||
|
|
if (isConstantNode(_arg) && (isZero(_arg.value) || equal(_arg.value, 1))) {
|
||
|
|
return createConstantNode(0);
|
||
|
|
} // d/dx(c^f(x)) = c^f(x)*ln(c)*f'(x)
|
||
|
|
|
||
|
|
|
||
|
|
return new OperatorNode('*', 'multiply', [node, new OperatorNode('*', 'multiply', [new FunctionNode('log', [_arg.clone()]), _derivative(_arg2.clone(), constNodes)])]);
|
||
|
|
}
|
||
|
|
|
||
|
|
if (constNodes[_arg2] !== undefined) {
|
||
|
|
if (isConstantNode(_arg2)) {
|
||
|
|
// If is secretly constant; f(x)^0 = 1 -> d/dx(1) = 0
|
||
|
|
if (isZero(_arg2.value)) {
|
||
|
|
return createConstantNode(0);
|
||
|
|
} // Ignore exponent; f(x)^1 = f(x)
|
||
|
|
|
||
|
|
|
||
|
|
if (equal(_arg2.value, 1)) {
|
||
|
|
return _derivative(_arg, constNodes);
|
||
|
|
}
|
||
|
|
} // Elementary Power Rule, d/dx(f(x)^c) = c*f'(x)*f(x)^(c-1)
|
||
|
|
|
||
|
|
|
||
|
|
var powMinusOne = new OperatorNode('^', 'pow', [_arg.clone(), new OperatorNode('-', 'subtract', [_arg2, createConstantNode(1)])]);
|
||
|
|
return new OperatorNode('*', 'multiply', [_arg2.clone(), new OperatorNode('*', 'multiply', [_derivative(_arg, constNodes), powMinusOne])]);
|
||
|
|
} // Functional Power Rule, d/dx(f^g) = f^g*[f'*(g/f) + g'ln(f)]
|
||
|
|
|
||
|
|
|
||
|
|
return new OperatorNode('*', 'multiply', [new OperatorNode('^', 'pow', [_arg.clone(), _arg2.clone()]), new OperatorNode('+', 'add', [new OperatorNode('*', 'multiply', [_derivative(_arg, constNodes), new OperatorNode('/', 'divide', [_arg2.clone(), _arg.clone()])]), new OperatorNode('*', 'multiply', [_derivative(_arg2, constNodes), new FunctionNode('log', [_arg.clone()])])])]);
|
||
|
|
}
|
||
|
|
|
||
|
|
throw new Error('Operator "' + node.op + '" is not supported by derivative, or a wrong number of arguments is passed');
|
||
|
|
}
|
||
|
|
});
|
||
|
|
/**
|
||
|
|
* Ensures the number of arguments for a function are correct,
|
||
|
|
* and will throw an error otherwise.
|
||
|
|
*
|
||
|
|
* @param {FunctionNode} node
|
||
|
|
*/
|
||
|
|
|
||
|
|
|
||
|
|
function funcArgsCheck(node) {
|
||
|
|
// TODO add min, max etc
|
||
|
|
if ((node.name === 'log' || node.name === 'nthRoot' || node.name === 'pow') && node.args.length === 2) {
|
||
|
|
return;
|
||
|
|
} // There should be an incorrect number of arguments if we reach here
|
||
|
|
// Change all args to constants to avoid unidentified
|
||
|
|
// symbol error when compiling function
|
||
|
|
|
||
|
|
|
||
|
|
for (var i = 0; i < node.args.length; ++i) {
|
||
|
|
node.args[i] = createConstantNode(0);
|
||
|
|
}
|
||
|
|
|
||
|
|
node.compile().evaluate();
|
||
|
|
throw new Error('Expected TypeError, but none found');
|
||
|
|
}
|
||
|
|
/**
|
||
|
|
* Helper function to create a constant node with a specific type
|
||
|
|
* (number, BigNumber, Fraction)
|
||
|
|
* @param {number} value
|
||
|
|
* @param {string} [valueType]
|
||
|
|
* @return {ConstantNode}
|
||
|
|
*/
|
||
|
|
|
||
|
|
|
||
|
|
function createConstantNode(value, valueType) {
|
||
|
|
return new ConstantNode(numeric(value, valueType || config.number));
|
||
|
|
}
|
||
|
|
|
||
|
|
return derivative;
|
||
|
|
});
|