# How do you find the derivative of y= 3^(2x+7)?

Apr 29, 2016

$\frac{\mathrm{dy}}{\mathrm{dx}} = {3}^{2 x + 7} \ln \left(9\right)$

#### Explanation:

Using the derivative $\frac{d}{\mathrm{dx}} {a}^{x} = {a}^{x} \ln \left(a\right)$ together with the chain rule, we can note that ${3}^{2 x + 7}$ = $f \left(g \left(x\right)\right)$ where $f \left(x\right) = {3}^{x}$ and $g \left(x\right) = 2 x + 7$ and apply the chain rule to obtain:

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{d}{\mathrm{dx}} {3}^{2 x + 7}$

$= \frac{d}{\mathrm{dx}} f \left(g \left(x\right)\right)$

$= f ' \left(g \left(x\right)\right) g ' \left(x\right)$

$= {3}^{2 x + 7} \ln \left(3\right) \cdot 2$

$= {3}^{2 x + 7} \ln \left(9\right)$