# What is the derivative of  e^(1/x)?

Mar 20, 2016

$\frac{d}{\mathrm{dx}} {e}^{\frac{1}{x}} = - {e}^{\frac{1}{x}} / {x}^{2}$
To find derivative of ${e}^{\frac{1}{x}}$, we use function of a function i.e. if $f \left(g \left(x\right)\right)$, $\frac{\mathrm{df}}{\mathrm{dx}} = \frac{\mathrm{df}}{\mathrm{dg}} \times \frac{\mathrm{dg}}{\mathrm{dx}}$
Hence $\frac{d}{\mathrm{dx}} {e}^{\frac{1}{x}}$ is equal to
${e}^{\frac{1}{x}} \times \frac{d}{\mathrm{dx}} \left(\frac{1}{x}\right) = {e}^{\frac{1}{x}} \times \left(- \frac{1}{x} ^ 2\right) = - {e}^{\frac{1}{x}} / {x}^{2}$