Derivative Calculator — Step-by-Step Solutions

The Derivative Calculator finds the derivative of any mathematical function with respect to a chosen variable. It supports polynomials, trigonometric functions, exponentials, logarithms, and their compositions.

Enter your function and variable, then choose the order of differentiation. The calculator uses symbolic differentiation to provide exact results displayed in standard mathematical notation.

f(x)

Variable

Your calculations will appear here

How to Use the Derivative Calculator

Enter a mathematical expression in the input field (e.g. x^3 + 2*x^2 - 5*x + 3).
Choose the variable to differentiate with respect to (default: x).
Select the order of differentiation (1st, 2nd, 3rd, etc.) and click Calculate.

Formula and Theory

The derivative of a function measures its instantaneous rate of change at any point. Geometrically, the derivative at a point gives the slope of the tangent line to the function's graph at that point.

Key derivative rules include: the power rule d/dx(x^n) = n*x^(n-1), the product rule d/dx(fg) = f'g + fg', the chain rule d/dx(f(g(x))) = f'(g(x)) * g'(x), and the quotient rule.

Worked Examples

Example 1: Power rule

Problem: Find the derivative of f(x) = 3x^4

Solution: Using the power rule: d/dx(3x^4) = 3 * 4 * x^3

Answer: f'(x) = 12x^3

Example 2: Trigonometric function

Problem: Find the derivative of f(x) = sin(x) * cos(x)

Solution: Using the product rule: cos(x)*cos(x) + sin(x)*(-sin(x)) = cos^2(x) - sin^2(x)

Answer: f'(x) = cos(2x)

Example 3: Chain rule

Problem: Find the derivative of f(x) = (2x + 1)^5

Solution: Outer: 5*(2x+1)^4, inner derivative: 2. Multiply: 5*2*(2x+1)^4

Answer: f'(x) = 10(2x + 1)^4

Example 4: Second derivative

Problem: Find the second derivative of f(x) = x^3 - 6x

Solution: f'(x) = 3x^2 - 6. f''(x) = 6x

Answer: f''(x) = 6x

Frequently Asked Questions

What is the derivative used for?

Derivatives are used to find rates of change, slopes of curves, velocities, optimisation (finding maxima and minima), and in physics, engineering, economics, and many other fields.

What is the chain rule?

The chain rule is used to differentiate composite functions. If y = f(g(x)), then dy/dx = f'(g(x)) * g'(x). It "chains" the derivatives of the outer and inner functions.

What is the difference between dy/dx and d/dx?

dy/dx represents the derivative of y with respect to x (a ratio of differentials). d/dx is the differentiation operator that acts on a function.

What does it mean if the derivative is zero?

A derivative of zero at a point means the function has a horizontal tangent there. This occurs at local maxima, local minima, or inflection points (critical points).