Derivative Rules: The Complete Cheat Sheet with Examples

Differentiation becomes mechanical once you know a handful of rules and recognise which one applies.

The core rules

Rule	Formula
Constant	$\frac{d}{dx}c=0$
Power	$\frac{d}{dx}{x}^n=n{x}^{n-1}$
Sum	$(f+g{)}^{\prime }=f^{\prime }+g^{\prime }$
Product	$(fg{)}^{\prime }=f^{\prime }g+f{g}^{\prime }$
Quotient	${\left(\frac{f}{g}\right)}^{\prime }=\frac{f^{\prime }g-f{g}^{\prime }}{g^2}$
Chain	$(f(g(x)){)}^{\prime }=f^{\prime }(g(x))g^{\prime }(x)$

Standard derivatives

$$\frac{d}{dx}{e}^x=e^x,\frac{d}{dx}\ln x=\frac{1}{x},\frac{d}{dx}\sin x=\cos x,\frac{d}{dx}\cos x=-\sin x$$

Worked examples

Product rule: $\frac{d}{dx}\left(x^2\sin x\right)=2x\sin x+x^2\cos x$.

Chain rule: $\frac{d}{dx}\left(e^{3x^2}\right)=e^{3x^2}\cdot 6x$.

Quotient rule: $\frac{d}{dx}\frac{x}{x+1}=\frac{1}{(x+1{)}^2}$.

How to decide which rule to use

Read the function from the outside in. If the outermost operation is a sum, split it. If it's a product or quotient, use those rules. If a function is applied to something other than plain $x$, you need the chain rule.

Practice with our Derivative Calculator, which names every rule it applies.