# What is the derivative of b^x where b is a constant?

Jan 11, 2016

$\frac{d}{\mathrm{dx}} \left[{b}^{x}\right] = {b}^{x} \cdot \ln b$

#### Explanation:

First, note that

${b}^{x} = {e}^{\ln} \left({b}^{x}\right) = {e}^{x \ln b}$

This allows us to differentiate the function using the chain rule:

$\frac{d}{\mathrm{dx}} \left[{e}^{x \ln b}\right] = {e}^{x \ln b} \cdot \frac{d}{\mathrm{dx}} \left[x \ln b\right]$

Just like $\frac{d}{\mathrm{dx}} \left[5 x\right] = 5$, $\frac{d}{\mathrm{dx}} \left[x \ln b\right] = \ln b$, since $\ln b$ will always be a constant.

This gives us a derivative of:

${e}^{x \ln b} \cdot \ln b$

Now, recall that ${e}^{x \ln b} = {b}^{x}$. This gives us our final, differentiated result:

$\frac{d}{\mathrm{dx}} \left[{b}^{x}\right] = {b}^{x} \cdot \ln b$