# What is the derivative of 5^(x^2)?

Nov 6, 2016

$\frac{d}{\mathrm{dx}} {5}^{{x}^{2}} = \left(2 \ln 5\right) x {5}^{{x}^{2}}$

#### Explanation:

Let $y = {5}^{{x}^{2}}$

Take Natural logs of both sides
$\ln y = \ln {5}^{{x}^{2}}$
$\therefore \ln y = {x}^{2} \ln 5$

Differentiate (implicitly) wrt $x$:
$\therefore \frac{1}{y} \frac{\mathrm{dy}}{\mathrm{dx}} = 2 x \ln 5$
$\therefore \frac{\mathrm{dy}}{\mathrm{dx}} = \left(2 \ln 5\right) x y$
$\therefore \frac{\mathrm{dy}}{\mathrm{dx}} = \left(2 \ln 5\right) x {5}^{{x}^{2}}$