∞Limitless

CALCULUS · DERIVATIVES

Derivative Rules

Power, product, quotient, and chain rules — the shortcuts that make differentiation fast.

18 min read 4 practice questions
Using the limit definition every time would be slow. These rules let you differentiate almost anything in a line or two.
ddx xn=nxn−1\frac{d}{dx}\,x^n = nx^{n-1}

Power rule

(fg)′=f′g+fg′(fg)′=f′g−fg′g2(fg)' = f'g + fg' \qquad\qquad \left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}

Product and quotient rules

ddx f(g(x))=f′(g(x))⋅g′(x)\frac{d}{dx}\,f(g(x)) = f'(g(x)) \cdot g'(x)

Chain rule: derivative of the outside × derivative of the inside

Worked example · The chain rule

Differentiate y=(3x2+1)5y = (3x^2 + 1)^5.
  1. 1
    Outside function: u5u^5, with u=3x2+1u = 3x^2 + 1.
  2. 2
    Derivative of the outside: 5u4=5(3x2+1)45u^4 = 5(3x^2 + 1)^4.
  3. 3
    Derivative of the inside: 6x6x.
  4. 4
    Multiply: y′=30x(3x2+1)4y' = 30x(3x^2 + 1)^4.
Common derivatives to memorize: (sin⁡x)′=cos⁡x(\sin x)' = \cos x, (cos⁡x)′=−sin⁡x(\cos x)' = -\sin x, (ex)′=ex(e^x)' = e^x, (ln⁡x)′=1x(\ln x)' = \frac{1}{x}.

CHECK YOUR UNDERSTANDING

Practice

Answer each question, then check.

1
Find ddx(4x3−5x+2)\frac{d}{dx}\left(4x^3 - 5x + 2\right).
2
If f(x)=x4f(x) = x^4, evaluate f′(2)f'(2).
3
Differentiate y=sin⁡(5x)y = \sin(5x).
4
Differentiate y=x2exy = x^2 e^x.

Numeric answers accept decimals or fractions like 3/4.