Derivative — Examples & Problems

Definition

The derivative represents the slope of the tangent line:

f'(x_0)=\lim_{h\to0}\frac{f(x_0+h)-f(x_0)}{h}

For finding local maximum and minimum, consider:

f'(x)=0 \qquad\text{or}\qquad f'(x)\text{ does not exist}

Basic derivative rules

(c)'=0

(x^n)'=nx^{n-1}

(cf(x))'=cf'(x)

(f(x)+g(x))'=f'(x)+g'(x)

(f(x)g(x))'=f'(x)g(x)+f(x)g'(x)

\left(\frac{f(x)}{g(x)}\right)' = \frac{f'(x)g(x)-f(x)g'(x)}{g(x)^2}

Chain Rule

(f(g(x)))'=f'(g(x))\cdot g'(x)

Common derivatives

(x^n)'=nx^{n-1}

(u^n)'=nu^{n-1}u'

(e^x)'=e^x

(\ln x)'=\frac{1}{x}

(\sin x)'=\cos x

(\cos x)'=-\sin x

(\tan x)'=1+\tan^2x

(\cot x)'=-(1+\cot^2x)

If:

f'(x)>0 \quad\Rightarrow\quad f(x)\text{ increases}

f'(x)<0 \quad\Rightarrow\quad f(x)\text{ decreases}