# Math 22 Differentiation

## The Constant Rule

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


Example: Find derivative of

1) ${\displaystyle f(x)=5}$

Solution:
${\displaystyle f'(x)=0}$

2) ${\displaystyle f(x)=\pi }$

Solution:
${\displaystyle f'(x)=0}$

3) ${\displaystyle f(x)=e^{2}}$

Solution:
${\displaystyle f'(x)=0}$

## The Power Rule

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


Example: Find derivative of

1) ${\displaystyle f(x)=x^{5}}$

Solution:
${\displaystyle f'(x)=(5)x^{5-1}=5x^{4}}$

2) ${\displaystyle f(x)=x^{1000}}$

Solution:
${\displaystyle f'(x)=(1000)x^{1000-1}=1000x^{999}}$

3) ${\displaystyle f(x)={\frac {1}{x^{3}}}}$

Solution:
We rewrite ${\displaystyle f(x)={\frac {1}{x^{3}}}=x^{-3}}$, so
${\displaystyle f'(x)=(-3)x^{-3-1}=-3x^{-4}={\frac {-3}{x^{4}}}}$

## The Constant Multiple Rule

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


1) ${\displaystyle f(x)=10x^{5}}$

Solution:
${\displaystyle f'(x)=10{\frac {d}{dx}}(x^{5})=10(5x^{4})=50x^{4}}$

2) ${\displaystyle f(x)=3x^{1000}}$

Solution:
${\displaystyle f'(x)=3{\frac {d}{dx}}(x^{1}000)=3(1000)x^{1000-1}=3000x^{999}}$

## The Sum and Difference Rules

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

${\displaystyle {\frac {d}{dx}}[f(x)-g(x)]=f'(x)-g'(x)}$


## Notes

1) ${\displaystyle {\frac {d}{dx}}[x]=1}$ since ${\displaystyle x^{0}=1}$

2) The derivative of ${\displaystyle f}$ is a function that gives the slope of the graph of ${\displaystyle f}$ at a point ${\displaystyle (x,f(x))}$.

3) We usually denote ${\displaystyle {\frac {d}{dx}}f(x)}$ as ${\displaystyle f'(x)}$