# Derivative of exponential functions

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##### Intros
###### Lessons
1. $\frac{d}{{dx}}\;{2^x}$
$\frac{d}{{dx}}\;{2^{4{x^3}}}$
##### Examples
###### Lessons
1. $\frac{d}{{dx}}\;{3^{{5^{{x^2}}}}}$
1. $\frac{{d}}{{{d}x}}\;{{e}^x}$
$\frac{{d}}{{{d}x}}\;{{e}^{\sin x}}$
1. Differentiate:
$y = {tan\;}(\;\cos {{e}^{5{x^2}}}\;)$
1. $\frac{d}{{dx}}\;{x^5}$ VS. $\frac{d}{{dx}}\;{5^x}$
###### Topic Notes
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.
Differential Rules – Exponential Functions

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

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