**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

We’re jumping ahead here, but it’s interesting to note: 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, since the denominator would be $ 0$: $ \displaystyle \int{{{{x}^{n}}}}dx=\frac{{{{x}^{{n+1}}}}}{{n+1}}\,+C,\,\,n\ne 1$. Thus, when we try to integrate a function like $ \displaystyle 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)$.

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$. 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**.

**Natural Log Differentiation Rules:**

Here are derivatives 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$. The derivative is:

$ \displaystyle \begin{align}\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)}}\end{align}$.

We can also 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 a 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 ****differentiation**”). (**Note**: We can also use the method of taking the **ln** on both sides for **differentiating complicated problems** without logarithmic or exponential functions, such as $ \displaystyle y=\frac{{{{{\left( {2x+1} \right)}}^{3}}}}{{{{x}^{5}}\sqrt{{x+1}}}}$. See below for an example.)

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 problems where we take the **ln** of both sides. **Note that the first problem isn’t even a log or exponent problem, but we’ll take the ln of both sides to make it much easier to differentiate!**

Here’s one more that uses **Implicit Differentiation**:

## Derivative of *e*^{u}

^{u}

Yeah! This is actually the easiest function to differentiate, since $ \displaystyle \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: $ \displaystyle \frac{d}{{dx}}\left( {{{e}^{u}}} \right)={{e}^{u}}\frac{{du}}{{dx}}$.

Here are 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 Derivatives of Inverse Functions – you’re ready!**