In calculus, a derivative measures the instantaneous rate at which a function changes. It's essentially the slope of the tangent line to the function's graph at any given point. Derivatives are fundamental tools in understanding how quantities change in relation to one another, with applications spanning physics, economics, engineering, and many other fields.
The Power Rule
One of the most common rules for differentiation is the Power Rule. For a function of the form f(x) = ax^n, where 'a' and 'n' are constants, its derivative f'(x) is given by f'(x) = n * ax^(n-1).
For a polynomial function, like f(x) = 3x^4 + 2x^2 - 5x + 7, we apply the Power Rule to each term individually. The derivative of a constant term (like 7) is always zero. So, the derivative f'(x) would be:
Derivative of 3x^4 is 4 * 3x^(4-1) = 12x^3
Derivative of 2x^2 is 2 * 2x^(2-1) = 4x^1 = 4x
Derivative of -5x (which is -5x^1) is 1 * -5x^(1-1) = -5x^0 = -5
Derivative of 7 is 0
Combining these, the derivative of f(x) = 3x^4 + 2x^2 - 5x + 7 is f'(x) = 12x^3 + 4x - 5.
Use Cases for Derivatives
Physics: Velocity is the derivative of position with respect to time. Acceleration is the derivative of velocity.
Economics: Marginal cost and marginal revenue are derivatives of total cost and total revenue functions, respectively, indicating the cost or revenue of producing one additional unit.
Optimization: Finding maximum or minimum values of functions (e.g., maximizing profit, minimizing cost) often involves finding where the derivative is zero.
Graphing: Derivatives help determine the slope, increasing/decreasing intervals, and concavity of a function's graph.
Limitations of this Calculator
This calculator is designed to handle basic polynomial functions and common operations like addition, subtraction, multiplication, and exponentiation (using '^'). It may not correctly interpret complex trigonometric, logarithmic, or exponential functions, or functions involving implicit differentiation or symbolic integration. For advanced calculus problems, specialized software is recommended.
function calculateDerivative() {
var functionStr = document.getElementById("functionInput").value;
var variable = document.getElementById("variable").value.trim() || 'x'; // Default to 'x' if empty
if (!functionStr) {
document.getElementById("result").textContent = "Please enter a function.";
return;
}
var resultDiv = document.getElementById("result");
var derivative = "";
try {
// Very basic parser and differentiator – handles polynomials and simple powers
// This is a simplified approach for demonstration and won't handle all cases.
// For robust symbolic differentiation, a dedicated library would be needed.
// Normalize function string: replace spaces, ensure variable is used consistently
functionStr = functionStr.replace(/\s+/g, ");
var regexVar = new RegExp(variable, 'g');
functionStr = functionStr.replace(/x/g, variable); // Replace all 'x' with the specified variable
// Split into terms
var terms = [];
var currentTerm = "";
var operator = '+';
for (var i = 0; i < functionStr.length; i++) {
var char = functionStr[i];
if (char === '+' || char === '-') {
if (currentTerm) {
terms.push({ term: currentTerm, operator: operator });
}
currentTerm = "";
operator = char;
} else {
currentTerm += char;
}
}
if (currentTerm) {
terms.push({ term: currentTerm, operator: operator });
}
var derivativeTerms = [];
for (var j = 0; j 1 ? parts[1] : ";
// Coefficient parsing
if (coeffStr === "") { // e.g., x^2 or just x
coefficient = 1;
} else if (coeffStr === "-") { // e.g., -x^2 or -x
coefficient = -1;
} else {
coefficient = parseFloat(coeffStr);
if (isNaN(coefficient)) {
throw new Error("Invalid coefficient in term: " + term);
}
}
// Exponent parsing
if (expStr === "") { // e.g., 5x (exponent is 1)
exponent = 1;
} else if (expStr.startsWith("^")) {
var expVal = expStr.substring(1);
if (expVal === "") { // e.g., 5x^
throw new Error("Incomplete exponent in term: " + term);
}
exponent = parseFloat(expVal);
if (isNaN(exponent)) {
throw new Error("Invalid exponent in term: " + term);
}
} else {
// This case might occur if the variable is followed by something unexpected
throw new Error("Unexpected format after variable in term: " + term);
}
// Apply Power Rule: n * ax^(n-1)
var newCoefficient = coefficient * exponent;
var newExponent = exponent – 1;
var derivativeTermStr = "";
if (newCoefficient !== 0) {
if (newExponent === 0) { // Result is a constant
derivativeTermStr = newCoefficient.toString();
} else if (newExponent === 1) { // Result is linear
if (newCoefficient === 1) derivativeTermStr = variable;
else if (newCoefficient === -1) derivativeTermStr = "-" + variable;
else derivativeTermStr = newCoefficient + variable;
} else { // Result has a variable exponent
if (newCoefficient === 1) derivativeTermStr = variable + "^" + newExponent;
else if (newCoefficient === -1) derivativeTermStr = "-" + variable + "^" + newExponent;
else derivativeTermStr = newCoefficient + variable + "^" + newExponent;
}
}
if (derivativeTermStr) {
derivativeTerms.push({ term: derivativeTermStr, operator: termOperator });
}
}
// Construct the final derivative string
if (derivativeTerms.length === 0) {
derivative = "0";
} else {
var finalDerivative = "";
for (var k = 0; k < derivativeTerms.length; k++) {
var dTermInfo = derivativeTerms[k];
if (k === 0) {
if (dTermInfo.operator === '-') {
finalDerivative += "-" + dTermInfo.term;
} else {
finalDerivative += dTermInfo.term;
}
} else {
if (dTermInfo.operator === '-') {
finalDerivative += " – " + dTermInfo.term;
} else {
finalDerivative += " + " + dTermInfo.term;
}
}
}
derivative = finalDerivative;
}
resultDiv.textContent = "Derivative: " + derivative;
} catch (error) {
resultDiv.textContent = "Error: " + error.message;
console.error("Calculation error:", error);
}
}