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

Notes

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

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

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

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