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 eu
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!