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Derivatives Of Exponential Functions [Guide] by dkmathstats

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hive-163521 · @dkmathstats ·
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Derivatives Of Exponential Functions [Guide]
Hi there. In this math post I cover the derivatives of exponential functions.

Math images rendered in LaTeX with QuickLatex.com.

<center><img src="https://cdn.pixabay.com/photo/2020/02/17/09/15/board-4855963_960_720.jpg" /></center>
<center><a href="https://cdn.pixabay.com/photo/2020/02/17/09/15/board-4855963_960_720.jpg">Image Source</a></center>

&nbsp;

## Quick Review Of Exponential Functions
---

An exponential function has a number base to the power of an exponent that is a variable. One example of an exponential function is five to the power of x.

<center>https://quicklatex.com/cache3/ea/ql_5c14bc188bc4bd9c1b264f3fe8dc7aea_l3.png</center>

&nbsp;

When x is equal to 1 you have the output `f(x)` equal to 5. With x equal to two you have five to the power of two which is 25. The input is the exponent. 

&nbsp;

<center><img src="https://cdn.pixabay.com/photo/2021/11/10/18/44/mathematics-6784603_960_720.jpg" /></center>
<center><a href="https://cdn.pixabay.com/photo/2021/11/10/18/44/mathematics-6784603_960_720.jpg">Image Source</a></center>

## Obtaining The Derivative Of Exponential Functions
---

The derivative of the exponential function is the slope of the exponential function at a given point x. Slope can also be viewed as rate of change. 

For the derivative of the exponential function, here is the general method. Note that b is a number base and the function `g(x)` is another function of x in the exponent.

<center>https://quicklatex.com/cache3/2e/ql_68e924773045aa475af182e9107da42e_l3.png</center>

Step One: For the derivative. Take the natural logarithm of the base. This would be `ln(b)`.

<center>https://quicklatex.com/cache3/71/ql_d94fe52bb639b37cc5be1828352eaf71_l3.png</center>

Step Two: Multiply this natural logarithm with the original exponential function `f(x)`.

<center>https://quicklatex.com/cache3/b5/ql_1dd0d20ce120692ae96d9ad808bb26b5_l3.png</center>

Step Three: Multiply by the derivative of the exponent `g(x)`. You are multiplying by `g'(x)` via the Chain Rule.

<center>https://quicklatex.com/cache3/e1/ql_f677a75dabae70dc037f29e9d3df31e1_l3.png</center>

&nbsp;

This may look complicated but let's look at some examples.

<center><img src="https://cdn.pixabay.com/photo/2024/02/21/13/49/ai-generated-8587765_960_720.jpg" /></center>
<center><a href="https://cdn.pixabay.com/photo/2024/02/21/13/49/ai-generated-8587765_960_720.jpg">Image Source</a></center>


&nbsp;

## Example One
---

What is the derivative of two to the power of x?

<center>https://quicklatex.com/cache3/a7/ql_a35e897537cba8d22dbe1823ba7968a7_l3.png</center>

&nbsp;

Take the natural logarithm of the base two to start.

<center>https://quicklatex.com/cache3/0a/ql_46f8cab9d6c678e4152a889fe6f5da0a_l3.png</center>

&nbsp;

Multiply the natural logarithm by the original exponential. That is multiply by 2 to the power of x.

<center>https://quicklatex.com/cache3/08/ql_62b448d625dea869a6c26d29a45d9908_l3.png</center>

&nbsp;

From the Chain rule take the derivative of the exponent. The derivative of `x` is just 1. Multiply one with the above.

<center>https://quicklatex.com/cache3/cd/ql_3c6466898aea5f3f710defa96791cccd_l3.png</center>

&nbsp;

## Example Two
---

What is the derivative of five to the power of -x?

<center>https://quicklatex.com/cache3/9b/ql_56d4546723bcf30496fef68849eac99b_l3.png</center>

&nbsp;

First, take the natural logarithm of five.

<center>https://quicklatex.com/cache3/4c/ql_982058d30e475b438746c8ed79c3494c_l3.png</center>

&nbsp;

Next, copy the original exponential function and multiply this with `ln(5)`.

<center>https://quicklatex.com/cache3/29/ql_95c4a802a11e785c29959662ed878729_l3.png</center>

&nbsp;

From the chain rule multiply by the derivative of the exponent. The derivative of the exponent negative x is negative one.

<center>https://quicklatex.com/cache3/56/ql_c1380a79025e9e47537b6d9171140656_l3.png</center>

&nbsp;

## Example Three
---

Compute the derivative of ten to the power of x-cubed.


&nbsp;


<center>https://quicklatex.com/cache3/3c/ql_ef55de359773eb81ccccd0df8805313c_l3.png</center>

&nbsp;

Take the natural logarithm of ten and multiply by the original exponential function.

<center>https://quicklatex.com/cache3/ce/ql_9140a72114b80a907120c7bf45f032ce_l3.png</center>

&nbsp;

Then multiply by the derivative of the exponent. The exponent here is x cubed so the derivative of this 3 x-squared. Multiply this 3 x-squared with the other components.


<center>https://quicklatex.com/cache3/1f/ql_1cd4e6846289a8a078eacba72e7caa1f_l3.png</center>

&nbsp;

<center><img src="https://cdn.pixabay.com/photo/2020/09/24/16/50/board-5599231_960_720.png" /></center>
<center><a href="https://cdn.pixabay.com/photo/2020/09/24/16/50/board-5599231_960_720.png">Image Source</a></center>

<center>Thank you for reading.</center>

Posted using [STEMGeeks](https://stemgeeks.net/@dkmathstats/derivatives-of-exponential-functions-guide)
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