# Math 22 Differentiation

## The Constant Rule

 The derivative of a constant function is zero. That is, ${\frac {d}{dx}}[c]=0$ where
$c$ is a constant


Example: Find derivative of

1) $f(x)=5$ Solution:
$f'(x)=0$ 2) $f(x)=\pi$ Solution:
$f'(x)=0$ 3) $f(x)=e^{2}$ Solution:
$f'(x)=0$ ## The Power Rule

 ${\frac {d}{dx}}[x^{n}]=nx^{n-1}$ for $n$ is a real number.


Example: Find derivative of

1) $f(x)=x^{5}$ Solution:
$f'(x)=(5)x^{5-1}=5x^{4}$ 2) $f(x)=x^{1000}$ Solution:
$f'(x)=(1000)x^{1000-1}=1000x^{999}$ 3) $f(x)={\frac {1}{x^{3}}}$ Solution:
We rewrite $f(x)={\frac {1}{x^{3}}}=x^{-3}$ , so
$f'(x)=(-3)x^{-3-1}=-3x^{-4}={\frac {-3}{x^{4}}}$ ## The Constant Multiple Rule

 If $f$ is a differentiable function of $x$ and $c$ is a real
number, then ${\frac {d}{dx}}[cf(x)]=cf'(x)$ for $c$ is a constant.


1) $f(x)=10x^{5}$ Solution:
$f'(x)=10{\frac {d}{dx}}(x^{5})=10(5x^{4})=50x^{4}$ 2) $f(x)=3x^{1000}$ Solution:
$f'(x)=3{\frac {d}{dx}}(x^{1}000)=3(1000)x^{1000-1}=3000x^{999}$ ## The Sum and Difference Rules

 The derivative of the sum or difference of two differentiable functions is the sum or difference
of their derivatives.
${\frac {d}{dx}}[f(x)+g(x)]=f'(x)+g'(x)$ ${\frac {d}{dx}}[f(x)-g(x)]=f'(x)-g'(x)$ 