What is the chain rule?

The chain rule is used for composite functions: d/dx[f(g(x))] = f'(g(x))·g'(x). Differentiate the outer function, keep the inner function unchanged, then multiply by the derivative of the inner function. For example, d/dx[sin(2x)] = cos(2x)·2 = 2cos(2x).

What is the derivative of sin(x)?

The derivative of sin(x) is cos(x). Similarly, d/dx[cos(x)] = -sin(x), d/dx[tan(x)] = sec²(x). These are fundamental trigonometric derivatives that should be memorized.

What is the derivative of e^x?

The derivative of e^x is e^x — it's the only function that is its own derivative. This property makes e^x essential in calculus. For e^(kx), use the chain rule: d/dx[e^(kx)] = k·e^(kx).

What is the derivative of ln(x)?

The derivative of ln(x) is 1/x. For ln(f(x)), use the chain rule: d/dx[ln(f(x))] = f'(x)/f(x). For example, d/dx[ln(x²)] = 2x/x² = 2/x.

When do you use the product rule vs quotient rule?

Use the product rule for f(x)·g(x): (fg)' = f'g + fg'. Use the quotient rule for f(x)/g(x): (f/g)' = (f'g - fg')/g². The quotient rule can be remembered as 'low d-high minus high d-low over low squared.'

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Derivative Calculator

Calculate derivatives of functions with detailed step-by-step solutions. Learn differentiation rules including power, product, quotient, and chain rules.

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Calculate Derivatives Step-by-Step

Find derivatives of any function with our free derivative calculator. Get detailed step-by-step solutions showing which differentiation rules to apply. Perfect for calculus students learning derivatives, homework help, and exam preparation.

What is a Derivative?

A derivative measures the instantaneous rate of change of a function. Written as f'(x) or dy/dx, it represents the slope of the tangent line to the function's graph at any point. Derivatives are fundamental to calculus, used in physics for velocity and acceleration, in economics for marginal analysis, and throughout science and engineering.

Derivative Notation

f'(x) = lim(h→0) [f(x+h) - f(x)]/h

Why Use This Calculator

Step-by-Step Solutions

See exactly which rules are applied at each step of the differentiation process.

Learn All Differentiation Rules

Master the power, product, quotient, and chain rules through worked examples.

Comprehensive Function Library

Practice with polynomials, trigonometric, exponential, and logarithmic functions.

How to Find a Derivative

Applications of Derivatives

Finding Rates of Change

Calculate velocity from position, acceleration from velocity, or marginal cost from total cost.

Optimization Problems

Find maximum and minimum values by setting the derivative equal to zero.

Curve Sketching

Determine where functions increase, decrease, and have inflection points.

Physics and Engineering

Model motion, electrical circuits, heat transfer, and countless physical phenomena.

Frequently Asked Questions

The power rule states that d/dx[x^n] = n·x^(n-1). Multiply by the exponent and reduce the exponent by 1. For example, d/dx[x³] = 3x². This works for any real number exponent, including negative and fractional powers.

What is a Derivative?

Derivative Notation

f'(x) = lim(h→0) [f(x+h) - f(x)]/h