Derivatives Of Exponential Functions

Table of Contents

  1. Exponential Functions: A Brief Review
  2. The Derivative of An Exponential Function
  3. Examples


Exponential Functions: A Brief Review

An exponential function is a function of the form:

\displaystyle a^{x}

where a is a non-zero number and x is a variable.

One should be careful and make the distinction between an exponential function such as 2^x versus a polynomial such as x^2 which is a variable to a numeric power/exponent.


The Derivative of An Exponential Function

If we are given f(x) = a^{g(x)} where g(x) is a different function of f(x). The derivative of f(x) is:

\displaystyle \begin{array} {lcl} f'(x) & = & \text{ln}(a) \cdot a^{g(x)} \cdot \dfrac{d}{dx} g(x) \\ & = & \text{ln}(a) \cdot a^{g(x)} \cdot g'(x) \end{array} \\

Note that this general formula does use a variation of the chain rule. Since the exponent is a function of x, we take the derivative of the exponent as well.

Given the more common case of g(x) = x in the exponent, the general case becomes:

\displaystyle \begin{array} {lcl} f'(x) & = & \text{ln}(a) \cdot a^{x} \cdot \dfrac{d}{dx} x \\ & = & \text{ln}(a) \cdot a^{x} \cdot (x)' \\ & = & \text{ln}(a) \cdot a^{x} \cdot 1 \\ & = & \text{ln}(a) \cdot a^{x} \\ \end{array} \\

If we are given e as the base such that we have f(x) = \text{e}^{g(x)}. The derivative f'(x) will be as follows:

\displaystyle \begin{array} {lcl} f'(x) & = & \text{ln}(a) \cdot \text{e}^{g(x)} \cdot \dfrac{d}{dx} g(x) \\ & = & \text{ln}(e) \cdot \text{e}^{x} \cdot g'(x) \\ & = & 1 \cdot \text{e}^{x} \cdot g'(x) \\ & = & \text{e}^{x} \cdot g'(x) \\ \end{array} \\

(Note that ln(e) = 1 as e^{1} = e.)

A special case is where f(x) = \text{e}^{x}

In this case we have a = e and g(x) = x. The derivative of \text{e}^{x} is simply \text{e}^{x}.


Examples

Example One: f(x) = \text{e}^{2x}

\displaystyle \begin{array} {lcl} f'(x) & = & \text{ln(e)} \cdot \text{e}^{2x} \cdot \dfrac{d}{dx} 2x \\ & = & 1 \cdot \text{e}^{2x} \cdot 2 \\ & = & 2 \cdot \text{e}^{2x} \\ \end{array}

Example Two: f(x) = 2 \cdot \text{e}^{x}

\displaystyle \begin{array} {lcl} f'(x) & = & 2 \cdot \dfrac{d}{dx} \text{e}^{x} \\ & = & 2 \cdot \text{e}^{x} \\ \end{array}

Example Three: f(x) = 5^{x}

\displaystyle \begin{array} {lcl} f'(x) & = & \text{ln(5)} \cdot 5^{x} \cdot \dfrac{d}{dx} x \\ & = & \text{ln(5)} \cdot 5^{x} \cdot 1 \\ & = & \text{ln(5)} \cdot 5^{x} \\ \end{array}

Example Four: f(x) = 10^{\text{sin}(x)}

\displaystyle \begin{array} {lcl} f'(x) & = & \text{ln(10)} \cdot 10^{\text{sin}(x)} \cdot \dfrac{d}{dx} \text{sin}(x) \\ & = & \text{ln(10)} \cdot 10^{\text{sin}(x)} \cdot \text{cos}(x)\\ \end{array}

The featured image is from http://www.emathhelp.net/images/calc/1_2_exponential_function.png.

Leave a Reply