Calculus guide

Derivative Rules: Power, Product, Quotient, and Chain Rules

Learn the four differentiation rules that solve most introductory derivative problems, with a complete worked example.

8 min readBeginnerUpdated 2026-07-12

What you will learn

Differentiate powers and sums
Recognize product and quotient structures
Apply the chain rule to composite functions

Concept 1

Start with the power rule

For a real exponent n, multiply by the exponent and reduce the exponent by one. Constants differentiate to zero, and sums can be differentiated term by term.

\frac{d}{dx}x^n=nx^{n-1}

Concept 2

Products and quotients need their own rules

Do not differentiate a product by multiplying the two derivatives. Keep one factor unchanged while differentiating the other, then reverse their roles.

For a quotient, square the denominator and preserve the order in the numerator.

(uv)'=u'v+uv',\qquad \left(\frac{u}{v}\right)'=\frac{u'v-uv'}{v^2}

Concept 3

Use the chain rule for functions inside functions

Differentiate the outside function first, leave the inside unchanged, and multiply by the derivative of the inside. Work from the outside inward when several layers are nested.

\frac{d}{dx}f(g(x))=f'(g(x))g'(x)

Worked example

Differentiate x² sin(x)

1Identify a product: u=x² and v=sin(x).
2Compute u'=2x and v'=cos(x).
3Apply u'v+uv' and simplify.

Answer

2x\sin(x)+x^2\cos(x)

Common mistakes

Multiplying u' and v' instead of using the product rule
Forgetting the inner derivative in the chain rule
Changing the order u'v−uv' in the quotient rule

Check your understanding

Try a problem with the Derivative Calculator

Open calculator

Products and quotients need their own rules

Do not differentiate a product by multiplying the two derivatives. Keep one factor unchanged while differentiating the other, then reverse their roles.

For a quotient, square the denominator and preserve the order in the numerator.

(uv)'=u'v+uv',\qquad \left(\frac{u}{v}\right)'=\frac{u'v-uv'}{v^2}