Derivatives

Definition of the derivative and basic rules.

Definition

The derivative of ff at xx is the limit of the difference quotient:

f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

Basic rules

RuleFormula
Powerddxxn=nxn−1\frac{d}{dx} x^n = n x^{n-1}
Product(fg)′=f′g+fg′(fg)' = f'g + fg'
Chain(f∘g)′(x)=f′(g(x)) g′(x)(f \circ g)'(x) = f'(g(x))\, g'(x)

Example

Differentiate f(x)=x2sin⁡xf(x) = x^2 \sin x using the product rule:

f′(x)=2xsin⁡x+x2cos⁡xf'(x) = 2x \sin x + x^2 \cos x