# Exercise ID138

Mathematical analysis → Derivation → Derivative of exponential functions
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Compute the derivative of the given function below.

$y={a}^{x}+{a}^{2x}+\frac{1}{2}{a}^{-x}$

$y={a}^{x}+{u}\left(x\right)+\frac{1}{2}{v}\left(x\right)\phantom{\rule{0ex}{0ex}}$ ${v}{=}{{a}}^{-x}{;}{}{g}{=}{-}{x}{;}{}{⇒}{v}{=}{{a}}^{g}{}{;}{}{g}{\text{'}}{=}{-}{1}\phantom{\rule{0ex}{0ex}}$ $y\text{'}={a}^{x}\mathrm{ln}a+{2}{{a}}^{2x}{l}{n}{}{a}{-}\frac{1}{2}{{a}}^{-x}{l}{n}{}{a}\phantom{\rule{0ex}{0ex}}$

$\frac{d\left({a}^{x}\right)}{dx}={a}^{x}·\mathrm{ln}a$

$\frac{d}{dx}\left[f\left(g\left(x\right)\right)\right]=\frac{df}{dg}·\frac{dg}{dx}=f{\text{'}}_{g}·g{\text{'}}_{x}$