This section covers:

**Introduction to Exponential and Logarithmic Differentiation and Integration****Differentiation of the Natural Logarithmic Function****General Logarithmic Differentiation****Inverses and Derivative of an Inverse****Derivative of***e*^{u}**More Practice**

**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**:

# 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 \(\ln x\).

**Natural Log Differentiation Rules:**

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

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.

# 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**:

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

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):

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

Here are more **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

**and solve for the “new**

*y***”. A function has an inverse function if it is**

*y***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 “

**” in the outside function, \(f\left( x \right)\).**

*x*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 “

*” for the particular “*

**y****” 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.*

**x**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:

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 *e*^{u}

^{u}

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:

**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!**