Enter a polynomial function and the variable with respect to which you want to differentiate it. This calculator supports basic polynomial functions using the power rule and sum/difference rules.
How to Use the Online Derivative Calculator
This tool helps you find the derivative of polynomial functions quickly. A derivative represents the instantaneous rate of change of a function with respect to one of its variables. In simpler terms, it tells you the slope of the tangent line to the function's graph at any given point.
What is a Derivative?
In calculus, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are fundamental to calculus and have wide-ranging applications in science, engineering, economics, and more.
Rate of Change: Derivatives are used to calculate rates of change, such as velocity (the derivative of position with respect to time) or acceleration (the derivative of velocity with respect to time).
Optimization: Finding the maximum or minimum values of a function (e.g., maximizing profit, minimizing cost) often involves setting the first derivative to zero.
Tangent Lines: The derivative at a point gives the slope of the tangent line to the function's graph at that point.
Supported Functions and Rules
This calculator is designed to handle polynomial functions and applies the following basic differentiation rules:
Power Rule: If \(f(x) = ax^n\), then \(f'(x) = anx^{n-1}\).
Constant Rule: If \(f(x) = c\) (where c is a constant), then \(f'(x) = 0\).
Constant Multiple Rule: If \(f(x) = c \cdot g(x)\), then \(f'(x) = c \cdot g'(x)\).
Sum/Difference Rule: If \(f(x) = g(x) \pm h(x)\), then \(f'(x) = g'(x) \pm h'(x)\).
Examples:
Derivative of \(5x^3\) is \(15x^2\).
Derivative of \(-2x\) is \(-2\).
Derivative of \(10\) is \(0\).
Derivative of \(4x^2 + 3x – 8\) is \(8x + 3\).
Limitations
Please note that this online calculator has certain limitations due to its client-side implementation. It currently supports:
Polynomial functions (e.g., \(ax^n + bx + c\)).
Differentiation with respect to a single variable (e.g., 'x').
It does not support:
Trigonometric functions (sin, cos, tan, etc.)
Logarithmic functions (ln, log)
Exponential functions (e^x)
Product Rule (e.g., \(f(x) \cdot g(x)\))
Quotient Rule (e.g., \(f(x) / g(x)\))
Chain Rule for complex nested functions (e.g., \((2x+1)^3\))
Functions with multiple variables (partial derivatives)
For more complex differentiation tasks, specialized mathematical software or more advanced symbolic calculators are recommended.
Example Calculation
Let's differentiate the function \(f(x) = 3x^2 – 5x + 7\) with respect to \(x\).
Input Function:3x^2 - 5x + 7
Input Variable:x
Click "Calculate Derivative"
Result:6x - 5
This is derived by applying the power rule to each term:
Derivative of \(3x^2\) is \(3 \cdot 2x^{2-1} = 6x\).
Derivative of \(-5x\) is \(-5 \cdot 1x^{1-1} = -5x^0 = -5\).
Derivative of \(7\) (a constant) is \(0\).
Combining these gives \(6x – 5\).
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function calculateDerivative() {
var functionStr = document.getElementById("functionInput").value;
var variable = document.getElementById("variableInput").value;
if (!functionStr) {
document.getElementById("result").innerHTML = "Error: Please enter a function.";
return;
}
if (!variable) {
document.getElementById("result").innerHTML = "Error: Please enter the differentiation variable.";
return;
}
if (variable.length !== 1 || !/[a-zA-Z]/.test(variable)) {
document.getElementById("result").innerHTML = "Error: Variable must be a single letter (e.g., 'x', 'y').";
return;
}
var result = differentiatePolynomial(functionStr, variable);
document.getElementById("result").innerHTML = "Derivative f'(" + variable + "): " + result;
}
function differentiatePolynomial(funcStr, variable) {
funcStr = funcStr.replace(/\s/g, "); // Remove all spaces
if (!funcStr) return "0"; // Derivative of empty string is 0 (or error, but 0 is safer for empty input after trim)
// Handle implicit coefficients like 'x^2' -> '1x^2' and '-x^2' -> '-1x^2'
// This regex ensures that 'x' becomes '1x' and 'x^n' becomes '1x^n'
funcStr = funcStr.replace(/([+-]|^)([a-zA-Z])(?!\^)/g, '$11$2'); // x -> 1x, +x -> +1x, -x -> -1x
funcStr = funcStr.replace(/([+-]|^)([a-zA-Z])(\^)/g, '$11$2$3'); // x^n -> 1x^n, +x^n -> +1x^n, -x^n -> -1x^n
// Prepend '+' if it doesn't start with '+' or '-'
if (!funcStr.startsWith('+') && !funcStr.startsWith('-')) {
funcStr = '+' + funcStr;
}
// Replace '-' with '+-' to easily split by '+'
funcStr = funcStr.replace(/-/g, '+-');
var terms = funcStr.split('+').filter(function(term) { return term !== "; });
var differentiatedTerms = [];
for (var i = 0; i < terms.length; i++) {
var term = terms[i];
var sign = 1;
if (term.startsWith('-')) {
sign = -1;
term = term.substring(1); // Remove the '-'
}
// Regex to match terms like '3x^2', '1x^3', '5x', '1x', '7'
// Group 1: coefficient (must be present after preprocessing)
// Group 2: variable (e.g., 'x')
// Group 3: exponent (optional)
var powerRuleMatch = term.match(/^(\d*\.?\d+)([a-zA-Z])(?:\^(\d*\.?\d+))?$/);
var constantMatch = term.match(/^(\d*\.?\d+)$/);
if (powerRuleMatch) {
var coeffStr = powerRuleMatch[1];
var varName = powerRuleMatch[2];
var expStr = powerRuleMatch[3];
if (varName !== variable) {
// If variable in term doesn't match differentiation variable, treat as constant
differentiatedTerms.push('0');
continue;
}
var coeff = parseFloat(coeffStr);
var exp = (expStr === undefined) ? 1 : parseFloat(expStr); // If no ^, exponent is 1
var newCoeff = coeff * exp * sign;
var newExp = exp – 1;
if (isNaN(newCoeff) || isNaN(newExp)) {
return "Error: Invalid number format in term '" + term + "'.";
}
if (newCoeff === 0) {
differentiatedTerms.push('0');
} else if (newExp === 0) {
differentiatedTerms.push(newCoeff.toString());
} else if (newExp === 1) {
differentiatedTerms.push(newCoeff + variable);
} else {
differentiatedTerms.push(newCoeff + variable + '^' + newExp);
}
} else if (constantMatch) {
// Constant term, derivative is 0
differentiatedTerms.push('0');
} else {
// Unsupported term format
return "Error: Unsupported function format or term '" + term + "'. This calculator supports polynomials like ax^n + bx + c.";
}
}
// Filter out '0' terms and combine
var finalResult = differentiatedTerms.filter(function(t) { return t !== '0'; });
if (finalResult.length === 0) {
return '0'; // Derivative of a constant or all terms became 0
}
var resultStr = finalResult.join('+');
// Clean up: replace '+-' with '-'
resultStr = resultStr.replace(/\+\-/g, '-');
// Clean up: remove leading '+' if present
if (resultStr.startsWith('+')) {
resultStr = resultStr.substring(1);
}
return resultStr;
}