# How do you find the derivative of e^(1/x) ?

Jun 23, 2016

$\frac{- {e}^{\frac{1}{x}}}{x} ^ 2$

#### Explanation:

Since the derivative of ${e}^{x}$ is just ${e}^{x}$, application of the chain rule to a composite function with ${e}^{x}$ as the outside function means that:

$\frac{d}{\mathrm{dx}} \left({e}^{f} \left(x\right)\right) = {e}^{f} \left(x\right) \cdot f ' \left(x\right)$

So, since the power of $e$ is $\frac{1}{x}$, we will multiply ${e}^{\frac{1}{x}}$ by the derivative of $\frac{1}{x}$.

Since $\frac{1}{x} = {x}^{-} 1$, its derivative is $- {x}^{-} 2 = - \frac{1}{x} ^ 2$.

Thus,

$\frac{d}{\mathrm{dx}} \left({e}^{\frac{1}{x}}\right) = {e}^{\frac{1}{x}} \cdot \left(- \frac{1}{x} ^ 2\right) = \frac{- {e}^{\frac{1}{x}}}{x} ^ 2$