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:

Exponential and Logarithmic Derivatives Exponential and Logarithmic Integrals Examples
\(\displaystyle \frac{d}{{dx}}\left( {\ln u} \right)=\frac{{{u}’}}{u}\) \(\displaystyle \int{{\frac{{{u}’}}{u}}}\,dx=\ln \left| u \right|\,+C\)

     \(\displaystyle \color{#800000}{{\frac{{d\left[ {\ln \left( {{{x}^{5}}-3} \right)} \right]}}{{dx}}}}=\frac{{{u}’}}{u}=\frac{{5{{x}^{4}}}}{{{{x}^{5}}-3}}\,\,\,\,\,\,\,\text{(}u={{x}^{5}}-3)\)

 

\(\displaystyle \begin{align}u&={{x}^{5}}-3\\du&=5{{x}^{4}}\,dx\\dx&=\frac{{du}}{{5{{x}^{4}}}}\end{align}\)          \(\require{cancel}  \begin{align}\color{#800000}{{\int{{\frac{{5{{x}^{4}}}}{{{{x}^{5}}-3}}}}\,dx}}&=\int{{\frac{{du}}{u}}}=\ln \left| u \right|+C\\&=\ln \left| {{{x}^{5}}-3} \right|+C\end{align}\)

\(\displaystyle \frac{d}{{dx}}\left( {{{{\log }}_{a}}u} \right)=\frac{{{u}’}}{{u\left( {\ln \,a} \right)}}\)

\(\displaystyle \int{{\frac{{{u}’}}{{u\ln a}}}}\,dx={{\log }_{a}}\left| u \right|+C\)

(Not a typical integration problem)

\(\displaystyle \color{#800000}{{\frac{d}{{dx}}\left[ {\log \left( {4x-1} \right)} \right]}}=\frac{d}{{dx}}\left[ {{{{\log }}_{{10}}}\left( {4x-1} \right)} \right]=\frac{4}{{\left( {4x-1} \right)\ln 10}}\)
\(\displaystyle \frac{d}{{dx}}\left( {{{e}^{u}}} \right)={{e}^{u}}{u}’\) \(\int{{{{e}^{u}}}}={{e}^{u}}+C\)

\(\displaystyle \color{#800000}{{\frac{{d\left( {{{e}^{{3x}}}} \right)}}{{dx}}}}={{e}^{{3x}}}\cdot 3=3{{e}^{{3x}}}\)

 

\(\begin{array}{l}u=3x\\du=3\,dx\end{array}\)             \(\displaystyle \color{#800000}{{\int{{3{{e}^{{3x}}}}}dx}}=3\int{{{{e}^{{3x}}}dx}}={{e}^{{3x}}}+C\)

\(\displaystyle \frac{d}{{dx}}\left( {{{a}^{u}}} \right)=\left( {\ln \,a} \right){{a}^{u}}{u}’\) \(\displaystyle \int{{{{a}^{u}}}}du=\left( {\frac{1}{{\ln a}}} \right){{a}^{u}}+C\)        \(\displaystyle \color{#800000}{{\frac{d}{{dx}}\left( {{{4}^{{3{{x}^{3}}}}}} \right)}}=\ln 4\left( {{{4}^{{3{{x}^{3}}}}}} \right)\left( {9{{x}^{2}}} \right)\)

 

\(\displaystyle \begin{align}u&=3{{x}^{3}}\\du&=9{{x}^{2}}\,dx\\dx&=\frac{{du}}{{9{{x}^{2}}}}\end{align}\)            \(\displaystyle \begin{array}{l}\color{#800000}{{\int{{\ln 4\left( {{{4}^{{3{{x}^{3}}}}}} \right)}}\left( {9{{x}^{2}}} \right)dx}}\\\,\,\,\,\,\,\,\,\,\,\,\,\,=\ln 4\int{{\left( {{{4}^{u}}} \right)}}\left( {\cancel{{9{{x}^{2}}}}} \right)\frac{{du}}{{\cancel{{9{{x}^{2}}}}}}\\\,\,\,\,\,\,\,\,\,\,\,\,\,=\ln 4\cdot \frac{1}{{\ln 4}}\cdot {{4}^{u}}+C={{4}^{{3{{x}^{3}}}}}+C\end{array}\)

\(\displaystyle \frac{d}{{dx}}\left[ {f{{{\left( x \right)}}^{{g\left( x \right)}}}} \right]\)

 

When we have a variable both in the base and the exponent, or if the function is really complicated, it’s best to take ln of both sides to take derivative, and use implicit integration. Then substitute \(y\) back in at the end.

Find Derivative: \(\displaystyle y=3{{x}^{x}}\)

 

\(\displaystyle \begin{align}\ln y&=\ln {{\left( {3x} \right)}^{x}};\ln y=x\ln \left( {3x} \right)\\\frac{1}{y}\left( {\frac{{dy}}{{dx}}} \right)&=x\cdot \frac{3}{x}+\ln \left( {3x} \right)\cdot 1\,\,\,\,\text{(product rule)}\\\,\frac{{dy}}{{dx}}&=\left[ {3+\ln \left( {3x} \right)} \right]\cdot y=3{{x}^{x}}\left[ {3+\ln \left( {3x} \right)} \right]\end{align}\)

Integrals of the Six Basic Trig Functions

\(\displaystyle \begin{array}{c}\int{{\sin u\,du=-\cos u\,\,+\,\,C}}\\\int{{\tan u\,du=-\ln \left| {\cos u} \right|\,\,+\,\,C\,}}\\\int{{\sec u\,du=\ln \left| {\sec u+\tan u} \right|\,\,+\,\,C}}\,\\\int{{\cos u\,du=\sin u\,\,+\,\,C}}\\\int{{\cot u\,du=\ln \left| {\sin u} \right|\,\,+\,\,C}}\\\int{{\csc u\,du=-\ln \left| {\csc u+\cot u} \right|\,\,+\,\,C}}\end{array}\)

\(\begin{align}u&=4\theta \\du&=4\,d\theta \end{align}\)          \(\begin{align}\color{#800000}{{\int{{\csc \left( {4\theta } \right)}}\,d\theta }}&=\frac{1}{4}\int{{\csc u\,du}}\\&=-\frac{1}{4}\ln \left| {\csc \left( {4x} \right)+\cot \left( {4x} \right)} \right|+C\end{align}\)

 

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: \(\displaystyle \int{{{{x}^{n}}}}dx=\frac{{{{x}^{{n+1}}}}}{{n+1}}\,+C,\,\,n\ne 1\).  So, 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 \(\ln \left( x \right)\).

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 \(\ln x\).

Logarithmic Properties

 

\(\begin{array}{c}\ln \left( {{{a}^{n}}} \right)=n\ln a\\\ln \left( {ab} \right)=\ln \left( a \right)+\ln \left( b \right)\\\ln \left( {\frac{a}{b}} \right)=\ln \left( a \right)-\ln \left( b \right)\\\ln \left( 1 \right)=0\\\ln \left( {{{e}^{x}}} \right)=x\end{array}\)               \(\displaystyle \begin{align}\text{Example:}\\\ln {{\left( {\frac{{{{x}^{4}}y}}{{\sqrt{z}}}} \right)}^{2}}&=2\ln \left( {\frac{{{{x}^{4}}y}}{{{{z}^{{\frac{1}{2}}}}}}} \right)=2\left[ {\ln \left( {{{x}^{4}}} \right)+\ln \left( y \right)-\ln \left( {{{z}^{{\frac{1}{2}}}}} \right)} \right]\\&=2\left[ {4\ln \left( x \right)+\ln \left( y \right)-\frac{1}{2}\ln \left( z \right)} \right]\\&=8\ln \left( x \right)+2\ln \left( y \right)-\ln \left( z \right)\end{align}\)

Natural Log Differentiation Rules:

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

Natural Log Differentiation

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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.

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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 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 differentation”).

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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):

Differentiate Exponential Function

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

Logarithmic Differentiation Problem Solution          

Find the derivative of the function:

 

\(\displaystyle f\left( x \right)=\log \left( {7x-2} \right)\)

\(\displaystyle \begin{align}\frac{{d\left( {{{{\log }}_{a}}u} \right)}}{{dx}}&=\frac{{{u}’}}{{u\ln a}}\\\frac{{d\left[ {{{{\log }}_{{10}}}\left( {7x-2} \right)} \right]}}{{dx}}&=\frac{7}{{\left( {7x-2} \right)\ln 10}}\end{align}\)

Find the derivative of the function:

 

\(\displaystyle f\left( x \right)=\log \left( {{{x}^{2}}\sqrt[3]{{\frac{{2x+1}}{{{{x}^{2}}-1}}}}} \right)\)

Expand first: \(\displaystyle f\left( x \right)=\log \left( {{{x}^{2}}\sqrt[3]{{\frac{{2x+1}}{{{{x}^{2}}-1}}}}} \right)=2\log \left( x \right)+\frac{1}{3}\log \left( {2x+1} \right)-\frac{1}{3}\log \left( {{{x}^{2}}-1} \right)\)

 

\(\displaystyle \begin{align}{f}’\left( {{{{\log }}_{a}}u} \right)&=\frac{{{u}’}}{{u\ln a}}=\frac{1}{{\ln 10}}\left[ {2\cdot \frac{1}{x}+\frac{1}{3}\left( {\frac{2}{{2x+1}}} \right)-\frac{1}{3}\left( {\frac{{2x}}{{{{x}^{2}}-1}}} \right)} \right]\\&=\frac{1}{{\ln 10}}\left[ {\frac{2}{x}+\frac{2}{{3\left( {2x+1} \right)}}-\frac{{2x}}{{3\left( {{{x}^{2}}-1} \right)}}} \right]\end{align}\)

 

(I could simplify with a common denominator, but I’m lazy 🙂 ).

Find the derivative of the function:

 

\(\displaystyle y=\frac{{5{{{\log }}_{2}}t}}{t}\)

Quotient Rule:   \(\require{cancel} \begin{align}{y}’&=5\left[ {\frac{{t\cdot \frac{{d\left( {{{{\log }}_{2}}t} \right)}}{{dt}}-{{{\log }}_{2}}t\cdot 1}}{{{{t}^{2}}}}} \right]=\frac{5}{{{{t}^{2}}}}\left( {\cancel{t}\cdot \frac{1}{{\cancel{t}\ln 2}}-{{{\log }}_{2}}t} \right)\\&=\frac{5}{{{{t}^{2}}}}\left( {\frac{1}{{\ln 2}}-\frac{{\ln t}}{{\ln 2}}} \right)=\frac{5}{{{{t}^{2}}\ln 2}}\left( {1-\ln t} \right)\end{align}\)

 

(I turned log into ln using Change of Base to simplify answer.)

 

Find an equation of the tangent line at the point \(\left( {3,\,1.5} \right)\):

 

\(\displaystyle y={{\log }_{2}}\sqrt{{{{x}^{2}}-1}}\)

\(\begin{align}y&={{\log }_{2}}{{\left( {{{x}^{2}}-1} \right)}^{{\frac{1}{2}}}}=\frac{1}{2}{{\log }_{2}}\left( {{{x}^{2}}-1} \right)\\{y}’&=\frac{{{u}’}}{{u\ln a}}=\frac{1}{2}\left[ {\frac{{2x}}{{\left( {{{x}^{2}}-1} \right)\ln 2}}} \right]=\frac{x}{{\left( {{{x}^{2}}-1} \right)\ln 2}}\end{align}\)

 

When \(x=3\), \(\displaystyle {y}’=\frac{3}{{\left( {{{3}^{2}}-1} \right)\ln \left( 2 \right)}}\approx .541\).

 

Tangent line through point \(\left( {3,1.5} \right)\): \(y-1.5=.541\left( {x-3} \right)\) or \(y=.541x-.123\).

 

 

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!

Differentiation Problem Solution          
Find the derivative of the function using logarithmic differentiation:

 

\(\displaystyle y=\frac{{{{{\left( {2x+1} \right)}}^{3}}}}{{{{x}^{5}}\sqrt{{x+1}}}}\)

Even though this isn’t a logarithmic or exponential problem, it’s a difficult problem to differentiate, so we can take the ln of each side to make it easier:

\(\displaystyle \begin{align}y&=\frac{{{{{\left( {2x+1} \right)}}^{3}}}}{{{{x}^{5}}\sqrt{{x+1}}}}\\\ln y&=\ln \left( {\frac{{{{{\left( {2x+1} \right)}}^{3}}}}{{{{x}^{5}}\sqrt{{x+1}}}}} \right)\\\ln y&=3\ln \left( {2x+1} \right)-5\ln x-\frac{1}{2}\ln \left( {x+1} \right)\\\frac{1}{y}\left( {\frac{{dy}}{{dx}}} \right)&=\frac{6}{{2x+1}}-\frac{5}{x}-\frac{1}{{2\left( {x+1} \right)}}\\&=\left( {\frac{6}{{2x+1}}-\frac{5}{x}-\frac{1}{{2x+2}}} \right)\cdot y\\&=\left( {\frac{6}{{2x+1}}-\frac{5}{x}-\frac{1}{{2x+2}}} \right)\cdot \frac{{{{{\left( {2x+1} \right)}}^{3}}}}{{{{x}^{5}}\sqrt{{x+1}}}}\end{align}\)

(Yes, we could probably simplify, but we’ll decide not to 😉)

Find the derivative of the function, by using logarithmic differentiation:

 

\(\displaystyle y={{x}^{{3x-1}}}\)

\(\displaystyle \begin{align}y&={{x}^{{3x-1}}}\\\ln y&=\ln {{x}^{{3x-1}}};\,\,\,\,\,\,\,\,\,\ln y=\left( {3x-1} \right)\ln x\\\frac{1}{y}\left( {\frac{{dy}}{{dx}}} \right)&=\left( {3x-1} \right)\frac{1}{x}+\ln \left( x \right)\cdot 3\,\,\,\,\\&\text{(implicit diff and product rule)}\\\frac{{dy}}{{dx}}&=\left[ {\left( {3x-1} \right)\frac{1}{x}+3\ln \left( x \right)} \right]\cdot y\\&={{x}^{{3x-1}}}\left[ {\frac{{3x-1}}{x}+3\ln \left( x \right)} \right]\end{align}\)
Find the derivative of the function, by using logarithmic differentiation:

 

\(f\left( x \right)={{\left( {3x+4} \right)}^{{{{x}^{2}}}}}\)

\(\displaystyle \begin{align}y={{\left( {3x+4} \right)}^{{{{x}^{2}}}}}\\\ln y&=\ln {{\left( {3x+4} \right)}^{{{{x}^{2}}}}};\,\,\,\,\,\,\ln y={{x}^{2}}\ln \left( {3x+4} \right)\\\frac{1}{y}\left( {\frac{{dy}}{{dx}}} \right)&={{x}^{2}}\cdot \frac{3}{{3x+4}}+\ln \left( {3x+4} \right)\cdot 2x\,\,\,\text{(product rule)}\\\frac{{dy}}{{dx}}&=\left[ {\frac{{3{{x}^{2}}}}{{3x+4}}+2x\cdot \ln \left( {3x+4} \right)} \right]\cdot y\\&={{\left( {3x+4} \right)}^{{{{x}^{2}}}}}\left[ {\frac{{3{{x}^{2}}}}{{3x+4}}+2x\ln \left( {3x+4} \right)} \right]\end{align}\)
Find the derivative of the function (don’t have to use logarithmic differentiation since the base isn’t a variable):

 

\(\displaystyle g\left( \theta \right)=\frac{{{{4}^{{\theta -2}}}}}{{\sin \left( {\pi \theta } \right)}}\)

\(\displaystyle \begin{align}{g}’\left( \theta \right)&=\frac{{\sin \left( {\pi \theta } \right)\cdot \ln 4\cdot {{4}^{{\theta -2}}}-{{4}^{{\theta -2}}}\cdot \cos \left( {\pi \theta } \right)\cdot \pi }}{{{{{\sin }}^{2}}\left( {\pi \theta } \right)}}\,\,\,\,\,\text{(quotient rule)}\\&=\frac{{{{4}^{{\theta -2}}}\left[ {\ln 4\cdot \sin \left( {\pi \theta } \right)-\pi \cos \left( {\pi \theta } \right)} \right]}}{{{{{\sin }}^{2}}\left( {\pi \theta } \right)}}\end{align}\)

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.

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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!

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