# Math 22 Derivatives of Exponential Functions

## Derivative of the Natural Exponential Function

 Let ${\displaystyle u}$ be a differentiable function of ${\displaystyle x}$. Then,
1.${\displaystyle {\frac {d}{dx}}[e^{x}]=e^{x}}$
2.${\displaystyle {\frac {d}{dx}}[e^{u}]=e^{u}{\frac {du}{dx}}}$


Exercises Differentiate each function:

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

Solution:
${\displaystyle f'(x)=2e^{2x}}$

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

Solution:
${\displaystyle f'(x)=6xe^{3x^{2}}}$

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

Solution:
${\displaystyle f'(x)=-2xe^{2x}}$

d) ${\displaystyle f(x)=4e^{-x}}$

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

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

Solution:
${\displaystyle f(x)={\frac {e^{x}-e^{-x}}{2}}={\frac {e^{x}}{2}}-{\frac {e^{-x}}{2}}={\frac {1}{2}}e^{x}-{\frac {1}{2}}e^{-x}}$
${\displaystyle f'(x)={\frac {1}{2}}e^{x}-{\frac {1}{2}}(-1)e^{-x}={\frac {1}{2}}e^{x}+{\frac {1}{2}}e^{-x}}$