Let u {\displaystyle u} be a differentiable function of x {\displaystyle x} . Then, 1. d d x [ e x ] = e x {\displaystyle {\frac {d}{dx}}[e^{x}]=e^{x}} 2. d d x [ e u ] = e u d u d x {\displaystyle {\frac {d}{dx}}[e^{u}]=e^{u}{\frac {du}{dx}}}
Exercises Differentiate each function:
a) f ( x ) = e 2 x {\displaystyle f(x)=e^{2x}}
b) f ( x ) = e 3 x 2 {\displaystyle f(x)=e^{3x^{2}}}
c) f ( x ) = e − x 2 {\displaystyle f(x)=e^{-x^{2}}}
d) f ( x ) = 4 e − x {\displaystyle f(x)=4e^{-x}}
e) f ( x ) = e x − e − x 2 {\displaystyle f(x)={\frac {e^{x}-e^{-x}}{2}}}
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![{\displaystyle {\frac {d}{dx}}[e^{x}]=e^{x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/783a58bf19ce36ca7781d7d31d78161a55406bff)
![{\displaystyle {\frac {d}{dx}}[e^{u}]=e^{u}{\frac {du}{dx}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9bd6499b9eb32d9110eb2dc6582eb380c15b2482)
