# Derivative of exponential functions

### Derivative of exponential functions

An exponential function is a function containing a numerical base with at least one variable in its exponent. In this section, we will learn how to differentiate exponential functions, including natural exponential functions and other composite functions that require the application of the Chain Rule.

#### Lessons

Differential Rules – Exponential Functions

$\frac{{d}}{{{d}x}}\;{c^x} = {c^x} \cdot \ln c$
$\frac{{d}}{{{d}x}}\;{c^{\left( {\;\;\;\;} \right)}} = {c^{\left( {\;\;\;\;} \right)}} \cdot \ln c \cdot \frac{{d}}{{{d}x}}\left( {\;\;\;\;} \right)$

$\frac{{d}}{{{d}x}}\;{{e}^x} = {{e}^x}$
$\frac{{d}}{{{d}x}}\;{{e}^{\left( {\;\;\;\;} \right)}} = {{e}^{\left( {\;\;\;\;} \right)}} \cdot \frac{{d}}{{{d}x}}\left( {\;\;\;\;} \right)$
• Introduction
$\frac{d}{{dx}}\;{2^x}$
$\frac{d}{{dx}}\;{2^{4{x^3}}}$

• 1.
$\frac{d}{{dx}}\;{3^{{5^{{x^2}}}}}$

• 2.
$\frac{{d}}{{{d}x}}\;{{e}^x}$
$\frac{{d}}{{{d}x}}\;{{e}^{\sin x}}$

• 3.
Differentiate:
$y = {tan\;}(\;\cos {{e}^{5{x^2}}}\;)$

• 4.
$\frac{d}{{dx}}\;{x^5}$ VS. $\frac{d}{{dx}}\;{5^x}$