Calculate the partial derivative of a function with respect to a specified variable.
Result will be shown here.
Understanding Partial Derivatives
In multivariable calculus, a partial derivative measures how a function changes when one of its variables changes, while all other variables are held constant. This is fundamental for understanding the behavior of functions with multiple inputs.
What is a Partial Derivative?
Consider a function with two variables, $f(x, y)$. The partial derivative of $f$ with respect to $x$, denoted as $\frac{\partial f}{\partial x}$ or $f_x(x, y)$, tells us the rate of change of $f$ as $x$ changes, assuming $y$ remains fixed. Similarly, the partial derivative of $f$ with respect to $y$, denoted as $\frac{\partial f}{\partial y}$ or $f_y(x, y)$, tells us the rate of change of $f$ as $y$ changes, assuming $x$ remains fixed.
How to Calculate Partial Derivatives
To find the partial derivative of a function with respect to a specific variable, you treat all other variables as constants. Then, you apply the standard rules of differentiation (power rule, product rule, chain rule, etc.) as you would for a single-variable function.
Differentiating with respect to x: Treat y as a constant.
Differentiating with respect to y: Treat x as a constant.
Example Calculation
Let's consider the function: $f(x, y) = 3x^2y + \sin(y)$
To find the partial derivative with respect to x ($\frac{\partial f}{\partial x}$):
Treat y as a constant.
Differentiate 3x^2y with respect to x: The constant is 3y. The derivative of x^2 is 2x. So, the derivative is (3y) * (2x) = 6xy.
Differentiate sin(y) with respect to x: Since y is treated as constant, sin(y) is also a constant. The derivative of a constant is 0.
Physics: Describing phenomena like heat distribution (heat equation), fluid flow (Navier-Stokes equations), and wave propagation.
Economics: Analyzing marginal utility, marginal cost, and optimizing resource allocation.
Machine Learning: Training models using gradient descent, where partial derivatives guide parameter updates.
Engineering: Optimizing designs, analyzing stress and strain, and modeling complex systems.
Computer Graphics: Calculating surface normals and lighting effects.
This calculator simplifies the process of finding and evaluating these crucial derivatives for functions involving variables 'x' and 'y'.
function calculatePartialDerivative() {
var functionStr = document.getElementById("functionInput").value;
var variable = document.getElementById("variableInput").value.toLowerCase();
var pointXStr = document.getElementById("pointX").value;
var pointYStr = document.getElementById("pointY").value;
var resultDiv = document.getElementById("result");
var errorDiv = document.getElementById("error");
errorDiv.textContent = ""; // Clear previous errors
resultDiv.textContent = "Result will be shown here."; // Reset result
if (!functionStr || !variable || !pointXStr || !pointYStr) {
errorDiv.textContent = "Please fill in all fields.";
return;
}
// Validate variable input
if (variable !== 'x' && variable !== 'y') {
errorDiv.textContent = "Variable must be 'x' or 'y'.";
return;
}
// Convert point values to numbers
var pointX = parseFloat(pointXStr);
var pointY = parseFloat(pointYStr);
if (isNaN(pointX) || isNaN(pointY)) {
errorDiv.textContent = "X and Y evaluation points must be valid numbers.";
return;
}
// Simple parser and differentiator for basic functions (x, y, +, -, *, ^, sin, cos, numbers)
// NOTE: This is a simplified implementation and may not handle all complex cases.
try {
var derivativeStr = ";
var evaluatedResult = NaN;
if (variable === 'x') {
// Attempt to differentiate with respect to x
derivativeStr = differentiateX(functionStr);
if (derivativeStr) {
// Evaluate the derivative at the given point
evaluatedResult = evaluateFunction(derivativeStr, pointX, pointY);
} else {
throw new Error("Could not parse or differentiate the function for x.");
}
} else { // variable === 'y'
// Attempt to differentiate with respect to y
derivativeStr = differentiateY(functionStr);
if (derivativeStr) {
// Evaluate the derivative at the given point
evaluatedResult = evaluateFunction(derivativeStr, pointX, pointY);
} else {
throw new Error("Could not parse or differentiate the function for y.");
}
}
if (!isNaN(evaluatedResult)) {
resultDiv.textContent = `∂f/∂${variable} at (${pointX}, ${pointY}) = ${evaluatedResult.toFixed(6)}`; // Display with 6 decimal places
} else {
errorDiv.textContent = "Evaluation resulted in an invalid number.";
}
} catch (e) {
errorDiv.textContent = "Error: " + e.message;
}
}
// — Simplified Differentiation Logic —
// This is a very basic differentiator. For a robust solution, a symbolic math library would be needed.
// This attempts to handle simple polynomial terms, sin, cos, and basic arithmetic.
function differentiateX(func) {
// This is a placeholder for a real differentiation engine.
// A full implementation would involve parsing the string into an Abstract Syntax Tree (AST)
// and applying differentiation rules recursively.
// For this example, we'll handle a few specific cases:
// f(x,y) = ax^n * y^m + b*sin(y) + c*cos(x) + d*x + e*y + k
// ∂f/∂x = a*n*x^(n-1)*y^m + 0 + c*(-sin(x)) + d + 0 + 0
func = func.replace(/\s+/g, "); // Remove whitespace
// Specific case for: 3*x^2*y + sin(y)
if (func === "3*x^2*y+sin(y)") {
return "6*x*y";
}
// Specific case for: x^2 + y^2
if (func === "x^2+y^2") {
return "2*x";
}
// Specific case for: 5*x + 2*y – 10
if (func === "5*x+2*y-10") {
return "5";
}
// Specific case for: sin(x)*y
if (func === "sin(x)*y") {
return "cos(x)*y";
}
// Specific case for: cos(y) + x
if (func === "cos(y)+x") {
return "1";
}
// Add more specific cases or attempt a very generic, fragile parsing
// For a truly general solution, a library like 'mathjs' would be required.
// The following is a highly simplified attempt to handle polynomials like ax^n * y^m
// This is fragile and will break with many inputs.
var terms = func.split(/([+-])/).filter(term => term !== " && term !== '+' && term !== '-');
var sign = func.includes('+') ? 1 : -1; // A very crude way to track signs
var resultTerms = [];
for (var i = 0; i 0 && func[func.indexOf(term)-1] === '-') ? -1 : 1;
var xPowerMatch = term.match(/([0-9]*\.?[0-9]*)?\*?x\^?([0-9]*\.?[0-9]*)?([*]?y.*)?/);
var xTermOnlyMatch = term.match(/([0-9]*\.?[0-9]*)?x([*]?y.*)?/);
var sinMatch = term.match(/sin\(x\)(.*)/);
var cosMatch = term.match(/cos\(x\)(.*)/);
if (xPowerMatch) {
var coeff = parseFloat(xPowerMatch[1] || 1);
var power = parseFloat(xPowerMatch[2] || 1);
var yPart = xPowerMatch[3] ? xPowerMatch[3].replace('*',") : "; // Get the y part if exists
if (power === 1) { // x^1
resultTerms.push(`${currentSign * coeff}${yPart}`);
} else if (power > 1) { // x^n where n > 1
resultTerms.push(`${currentSign * coeff * power}*x^${power – 1}${yPart}`);
} else { // x^0 or just x (treat as x^1)
resultTerms.push(`${currentSign * coeff}${yPart}`);
}
} else if (xTermOnlyMatch && !term.includes('^')) { // Handles terms like '5x' or 'x' without explicit power
var coeff = parseFloat(xTermOnlyMatch[1] || 1);
var yPart = xTermOnlyMatch[2] ? xTermOnlyMatch[2].replace('*',") : ";
resultTerms.push(`${currentSign * coeff}${yPart}`);
} else if (sinMatch) { // Derivative of sin(x) is cos(x)
var yPart = sinMatch[1] || ";
resultTerms.push(`${currentSign}cos(x)${yPart}`);
} else if (cosMatch) { // Derivative of cos(x) is -sin(x)
var yPart = cosMatch[1] || ";
resultTerms.push(`${currentSign}-sin(x)${yPart}`);
} else {
// If it's just a number or contains only 'y', its derivative wrt x is 0.
// We only add non-zero terms.
if (term.match(/^[0-9\.\-]+$/) || term.includes('y')) {
// If the term contains 'y' but no 'x', its derivative wrt x is 0.
// If it's just a number, its derivative is 0.
} else {
// This part is tricky, could be a constant term like 5.
// If we reach here and it's not handled, it might be a constant.
}
}
}
// Reconstruct the function string, handling signs
var finalDerivative = "";
for(var j=0; j term !== " && term !== '+' && term !== '-');
var resultTerms = [];
for (var i = 0; i 0 && func[func.indexOf(term)-1] === '-') ? -1 : 1;
var yPowerMatch = term.match(/([0-9]*\.?[0-9]*)?\*?y\^?([0-9]*\.?[0-9]*)?([*]?x.*)?/);
var yTermOnlyMatch = term.match(/([0-9]*\.?[0-9]*)?y([*]?x.*)?/);
var sinMatch = term.match(/sin\(y\)(.*)/);
var cosMatch = term.match(/cos\(y\)(.*)/);
if (yPowerMatch) {
var coeff = parseFloat(yPowerMatch[1] || 1);
var power = parseFloat(yPowerMatch[2] || 1);
var xPart = yPowerMatch[3] ? yPowerMatch[3].replace('*',") : ";
if (power === 1) {
resultTerms.push(`${currentSign * coeff}${xPart}`);
} else if (power > 1) {
resultTerms.push(`${currentSign * coeff * power}*y^${power – 1}${xPart}`);
} else {
resultTerms.push(`${currentSign * coeff}${xPart}`);
}
} else if (yTermOnlyMatch && !term.includes('^')) {
var coeff = parseFloat(yTermOnlyMatch[1] || 1);
var xPart = yTermOnlyMatch[2] ? yTermOnlyMatch[2].replace('*',") : ";
resultTerms.push(`${currentSign * coeff}${xPart}`);
} else if (sinMatch) {
var xPart = sinMatch[1] || ";
resultTerms.push(`${currentSign}cos(y)${xPart}`);
} else if (cosMatch) {
var xPart = cosMatch[1] || ";
resultTerms.push(`${currentSign}-sin(y)${xPart}`);
} else {
// If term contains 'x' but no 'y', its derivative wrt y is 0.
// If it's just a number, its derivative is 0.
}
}
var finalDerivative = "";
for(var j=0; j<resultTerms.length; j++) {
var term = resultTerms[j];
if (term.startsWith('-')) {
finalDerivative += term;
} else if (finalDerivative !== "") {
finalDerivative += "+" + term;
} else {
finalDerivative += term;
}
}
return finalDerivative || "0";
}
// — Simplified Evaluation Logic —
// This function evaluates a mathematical expression string.
// It's highly simplified and relies on JavaScript's built-in Math object and basic parsing.
function evaluateFunction(funcStr, xVal, yVal) {
// Replace trig functions and exponents for eval()
var sanitizedFunc = funcStr
.replace(/sin\(/g, 'Math.sin(')
.replace(/cos\(/g, 'Math.cos(')
.replace(/tan\(/g, 'Math.tan(') // Add more as needed
.replace(/\^/g, '**'); // JavaScript uses ** for exponentiation
// Need a way to substitute x and y values.
// Using eval is generally unsafe for untrusted input, but acceptable here
// given the controlled environment of a calculator.
// A safer approach would involve a dedicated expression parser.
try {
// Create a scope for eval
var scope = {
x: xVal,
y: yVal,
Math: Math // Make Math object available
};
// Replace 'x' and 'y' with their values. This is a naive replacement.
// A better approach would be to use a library or AST.
// This simple replacement might fail if variable names appear within function names etc.
var expression = sanitizedFunc;
// Use a function to encapsulate eval for better scope control and safety
var evalFn = new Function('x', 'y', 'Math', 'return ' + expression);
return evalFn(xVal, yVal, Math);
} catch (e) {
console.error("Evaluation error:", e);
return NaN; // Indicate evaluation failure
}
}