# What is the derivative of  3^x?

Jan 4, 2016

$\frac{d}{\mathrm{dx}} {3}^{x} = {3}^{x} \ln \left(3\right)$

#### Explanation:

An easy way of doing this is by using logarithmic differentiation.

To do this, we will use the following:

1. $\ln \left({a}^{x}\right) = x \ln \left(a\right)$
2. The chain rule
3. $\frac{d}{\mathrm{dx}} \ln \left(x\right) = \frac{1}{x}$
4. $\frac{d}{\mathrm{dx}} c x = c$

Let $y = {3}^{x}$

$\implies \ln \left(y\right) = \ln \left({3}^{x}\right) = x \ln \left(3\right)$

$\implies \frac{d}{\mathrm{dx}} \ln \left(y\right) = \frac{d}{\mathrm{dx}} x \ln \left(3\right)$

$\implies \frac{1}{y} \frac{\mathrm{dy}}{\mathrm{dx}} = \ln \left(3\right)$

$\implies \frac{\mathrm{dy}}{\mathrm{dx}} = y \ln \left(3\right) = {3}^{x} \ln \left(3\right)$

$\therefore \frac{d}{\mathrm{dx}} {3}^{x} = {3}^{x} \ln \left(3\right)$