Exponential and Logarithmic Differentiation

This section covers:

Exponential and Logarithmic Differentiation and Integration have a lot of practical applications and are handled a little differently than we are used to.  For a review of these functions, visit the Exponential Functions section and the Logarithmic Functions section

Note that we will address Exponential and Logarithmic Integration here in the Integration section.

Introduction to Exponential and Logarithmic Differentiation and Integration

Before getting started, here is a table of the most common Exponential and Logarithmic formulas for Differentiation and Integration:

Derivatives And Integrals Of Exponents And Logs

Differentiation of the Natural Logarithmic

When we learn the Power Rule for Integration here in the Antiderivatives and Integration section, we will notice that if n = –1, the rule doesn’t apply:  \(\int{{{{x}^{n}}}}dx=\frac{{{{x}^{{n+1}}}}}{{n+1}}\,+C,\,\,n\ne 1\).  So when we try to integrate a function like \(f\left( x \right)=\frac{1}{x}={{x}^{{-1}}}\), we have to do something “special”; namely learn that this integral is \(\ln \left( x \right)\).

Remember that \(\ln x\) is the same as \({{\log }_{e}}x\), where \(e\approx 2.718\) (“e” is Euler’s Number).  A log is the exponent raised to the base power (a) to get the argument (x) of the log (if “a” is missing, we assume it’s 10).

Here are some logarithmic properties that we learned here in the Logarithmic Functions section; note we could use \({{\log }_{a}}x\) instead of \(\ln x\).

Logarithm Properties

Natural Log Differentiation Rules:

Here’s how we take the derivative of natural logarithm functions:

Natural Log Differentiation

Here are some natural log (ln) differentiation problems.  Note that it’s typically easier to use the log properties to expand the function before differentiating.  Also note that you may not have to simplify the answers as much as shown.

Natural Log Differentiation Problems

General Logarithmic Differentiation

Logarithmic Differentiation gets a little trickier when we’re not dealing with natural logarithms.  Remember that from the change of base formula (for base a) that \(\displaystyle  {{\log }_{a}}x=\frac{{\log x}}{{\log a}}=\frac{{\ln x}}{{\ln a}}=\frac{1}{{\ln a}}\cdot \ln x\).   When we take the derivative of this, we get \(\displaystyle \frac{{d\left( {{{{\log }}_{a}}x} \right)}}{{dx}}=\frac{d}{{dx}}\left( {\frac{1}{{\ln a}}\cdot \ln x} \right)=\frac{1}{{\ln a}}\cdot \frac{d}{{dx}}\left( {\ln x} \right)=\frac{1}{{\ln a}}\cdot \frac{1}{x}=\frac{1}{{x\left( {\ln a} \right)}}\).

From these calculations, we can get the derivative of the exponential function \(y={{a}^{x}}\)  using implicit differentiation:

Derivative Of Exponential Function

Based on these derivations, here are the formulas for the derivative of the exponent and log functions:

Exponential And Logarithmic Differentiation Formulas

Note that for the last two problems above (exponential differentiation), we can just take the ln of each side and not worry about the “formula”.  In fact, when we have an variable such as x in the base and also the exponent (such as \(y=f{{\left( x \right)}^{{g\left( x \right)}}}\)), we need to take ln of both sides and use implicit differentiation to solve (called “logarithmic differentation”).

Here’s an example (using product rule):

Differentiate Exponential Function

Here are more logarithmic differentiation problems; note that typically want to expand logs before we integrate:

Log Differentiation Problems

Here are more exponential differentiation problems:

Exponential Differentiation Problems

Inverses and Derivative of an Inverse

Around the time you’re studying exponential and logarithmic differentiation and integration, you’ll probably learn how to get the derivative of an inverse function.  This is because some of the derivations of the exponential and log derivatives were a direct result of differentiating inverse functions.

We learned about inverse functions here in the Inverses of Functions section.  You get the inverse of a function if you switch the x and y and solve for the “new y”.  A function has an inverse function if it is one-to-one, which means it passes both vertical and horizontal line tests.

We can determine if a function is monotonic in an interval (and therefore has an inverse in that interval) if the derivative of that function is either greater than 0 (increasing) or less than 0 (decreasing) for that entire interval.

The derivative of an inverse function can be found the following way; note that \(f\left( {g\left( x \right)} \right)\) means a composite function, which means that we take the inside function, \(g\left( x \right)\), and put that in everywhere there’s an “x” in the outside function, \(f\left( x \right)\).

Derivative Of An Inverse Function Formula

What this says is if we have a function and want to find the derivative of the inverse of the function at a certain point “x”, we just find the “y” for the particular “x” in the original function, and use this value as the “x” in the derivative of this function. Then take the reciprocal of this number; this gives the derivative of the inverse of the original function at this point.

 Another way to explain this is “the derivative of \(f\left( x \right)\) at a point (a, b) is the reciprocal of the derivative of \({{f}^{{-1}}}\left( x \right)\) at point (b, a)”.  (We’ll do problems below).

Let’s first do some problems where we use the derivative to find out if a function has is strictly monotonic (has a strictly increasing or decreasing derivative) on its entire domain:

Function Monotonic

Here are some Derivative of the Inverse problems.  Some teachers may have you solve these using implicit differentiation, so I’m including that method, too:

Derivative Of An Inverse Function

Derivative of eu

Yeah!  This is actually the easiest function to differentiate, since \(\frac{d}{{dx}}\left( {{{e}^{x}}} \right)={{e}^{x}}\)!  I know; it’s strange, isn’t it?  This means that the slope of the graph of this function at any point is just equal to the y coordinate of that point.

When we have a function of x in the exponent, we just have to multiply by the derivative of this function: \(\frac{d}{{dx}}\left( {{{e}^{u}}} \right)={{e}^{u}}\frac{{du}}{{dx}}\).

Let’s do some problems:

Exponential Differentation Problems Understand these problems, and practice, practice, practice!

Click on Submit (the arrow to the right of the problem) to solve this problem. You can also type in more problems, or click on the 3 dots in the upper right hand corner to drill down for example problems.

If you click on “Tap to view steps”, you will go to the Mathway site, where you can register for the full version (steps included) of the software.  You can even get math worksheets.

You can also go to the Mathway site here, where you can register, or just use the software for free without the detailed solutions.  There is even a Mathway App for your mobile device.  Enjoy!

On to Antiderivatives and Indefinite Integrals  – you’re ready!