You CAN Ace Calculus  

17calculus > derivatives > exponential and logarithm  


Topics You Need To Understand For This Page
For the basic exponential and logarithm derivatives you do not need the chain rule. But we discuss it on this page. Each section is labeled. So if you have not studied the chain rule yet, you can read the sections that apply to you and then come back here once you have studied it. 
Calculus Main Topics
Derivatives 

Derivative Applications 
Single Variable Calculus 
MultiVariable Calculus 
Tools
math tools 

general learning tools 
additional tools 
Related Topics and Links
Exponential and Logarithm Derivatives 

on this page: ► exponential derivative ► exponential practice ► logarithm derivative ► logarithm practice 
The derivatives of the exponential and the logarithm are very different. However, since the functions themselves are closely related, we discuss both of them on this page. 
Exponential Derivative 
This is one of the easiest rules you will learn.
Basic Exponential Rule 
Exponential With Chain Rule  

\(\displaystyle{ \frac{d}{dt}[e^t] = e^t }\) 
\(\displaystyle{ \frac{d}{dt}[e^u] = e^u \frac{du}{dt} }\) 
Basic Exponential Rule 

\(\displaystyle{ \frac{d}{dt}[e^t] = e^t }\) 
Exponential With Chain Rule 
\(\displaystyle{ \frac{d}{dt}[e^u] = e^u \frac{du}{dt} }\) 
It looks like we didn't do anything here. However, the exponential function is the only function whose derivative is itself.
Okay, so what do you if you have a base other than \(e\)? The formula is fairly straightforward but let's derive from our rules of logarithms. Since we don't like to memorize formulas and we already know the logarithm rules, why not just derive it when we need it, since we don't use it very much?
So, let's convert \(y=a^x\), where \(a\) is a constant, into a form with \(e\).
\(\displaystyle{
\begin{array}{rcl}
y & = & a^x \\
\ln(y) & = & \ln(a^x) \\
& = & x\ln(a) \\
e^{\ln(y)} & = & e^{x\ln(a)} \\
y & = & e^{x\ln(a)}
\end{array}
}\)
So now when we take the derivative of \(y = a^x\), we can actually take the derivative of \(y=e^{x\ln(a)}\). Using the chain rule, we have \((a^x)' = (\ln(a))e^{x\ln(a)} \). Notice how didn't have to memorize this formula. We used the logarithm rules we already know.
This next video goes through all the explanation again. It is always good to get explanations from different sources since it will help you understand the material better.
PatrickJMT  Derivatives of Exponential Functions  
Here is an interesting video that shows how to get the equation for the derivative of \(f(x)=a^x\) another way. He shows that \(\displaystyle{f'(x)=\frac{d[a^x]}{dx}=a^x f'(0)}\). This is an interesting and unusual way to think about the derivative.
Dr Chris Tisdell  Derivative of exponentials  
So far, we've only been looking at equations with exponential functions. Here is a video discussing the graph, the derivative and the tangent line of three exponential functions. This helps you get more of an intuitive feel for this function and it's derivative.
MathTV  Some Natural Exponential Functions and Tangent Lines  
Exponential Practice 
Unless otherwise instructed, calculate the derivative of these functions.
exponential practice filters  

no chain rule  
chain rule required 
Basic Problems 
Practice 1 

\(\displaystyle{f(x)=(x^21)e^{x}}\) 
solution 
Practice 2 

\(\displaystyle{f(x)=xe^{\sqrt{x}}}\) 
solution 
Practice 3 

\(\displaystyle{f(x)=\frac{1e^{x}}{x}}\) 
solution 
Practice 4 

\(\displaystyle{y=e^x(x+x\sqrt{x})}\) 
solution 
Practice 5 

\(f(x)=4^x+3e^x+x^4\) 
solution 
Practice 6 

\(f(x)=e^xx^2\) 
solution 
Intermediate Problems 
Practice 8 

\(\displaystyle{g(x)=2e^{\cos(x)\sin(5x)}}\) 
solution 
Practice 9 

\(\displaystyle{f(x)=e^{x\sin(2x)}}\) 
solution 
Practice 10 

\(\displaystyle{f(t)=\cos\left(2^{\pi t}\right)}\) 
solution 
Practice 11 

\(\displaystyle{ y = \cos \left( \frac{1e^{2x}}{1+e^{2x}} \right) }\) 
solution 
Practice 12 

For what values of x does \(h(x)=5e^{5x}25x\) have negative derivatives? 
solution 
Logarithm Derivative 
The rule is not as simple as the exponential derivative but it is still very straightforward.
Basic Logarithm Rule 
Logarithm With Chain Rule  

\(\displaystyle{ \frac{d}{dt}[\ln(t)] = \frac{1}{t} }\) 
\(\displaystyle{ \frac{d}{dt}[\ln(u)] = \frac{1}{u}\frac{du}{dt} }\) 
Basic Logarithm Rule 

\(\displaystyle{ \frac{d}{dt}[\ln(t)] = \frac{1}{t} }\) 
Logarithm With Chain Rule 
\(\displaystyle{ \frac{d}{dt}[\ln(u)] = \frac{1}{u}\frac{du}{dt} }\) 
It is probably not clear just from the equation that the derivative of \(\ln(x)\) is \(1/x\). Here is a great video explaining, first intuitively, then from the limit, where this derivative comes from. Although he says you can stop the video after the intuitive explanation, watching the entire video will help you a lot (it's not very long).
MathTV  The Derivative of the Natural Log Function  
Many times, it helps to simplify a logarithmic expression before taking the derivative. Here are a few rules that should help you.
\( \ln(xy) = \ln(x) + \ln(y) \) 
\( \ln(x/y) = \ln(x)  \ln(y) \) 
\( \ln(x^y) = y \ln(x) \) 
Here is a short video clip that goes through these equations again.
PatrickJMT  Derivatives of Logarithmic Functions and Examples  
Before jumping into some practice problems, take a couple of minutes to watch this next video. It will help you see some common mistakes that you can avoid when taking the derivative of logarithm functions.
MathTV  Common Mistakes for Natural Logs and the Chain Rule  
Logarithm Practice 
Unless otherwise instructed, calculate the derivative of these functions.
logarithm practice filters  

no chain rule  
chain rule required 
Basic Problems 
Practice 13 

\(\displaystyle{f(x)=\frac{1}{\ln(x)}}\) 
solution 
Practice 14 

\(\displaystyle{f(x)=\sqrt{x}\ln(x)}\) 
solution 
Practice 15 

\(f(x)=\ln(x^2+10)\) 
solution 
Practice 16 

\(y=\ln(x^2+x)\) 
solution 
Practice 18 

\(h(x)=\ln(x^2+3x+4)\) 
solution 
Practice 19 

\(f(x)=\ln(5x^2+2x7)\) 
solution 
Practice 20 

\(\displaystyle{g(x)=\log_4(x^3+8x)}\) 
solution 
Practice 22 

\(\displaystyle{f(x)=\ln\left(x\sqrt{x^2+1}\right)}\) 
solution 
Practice 23 

\( y = \ln(x) + 2^x + \sin(x) \) 
solution 
Practice 24 

Calculate the first three derivatives of \( f(x)=\ln(2+3x) \). 
solution 
Intermediate Problems 
Practice 25 

\(\displaystyle{y=\frac{\ln x}{1+\ln(2x)}}\) 
solution 
Practice 26 

\(\displaystyle{f(x)=\ln\left[\frac{(2x+1)^5}{\sqrt{x^2+1}}\right]}\) 
solution 
Practice 27 

\(\displaystyle{f(x)=\ln\left[\frac{(2x+1)^3}{(3x1)^4}\right]}\) 
solution 
Practice 28 

\(\displaystyle{y=\sqrt[3]{\log_7(x)}}\) 
solution 
Practice 29 

\(y=\ln(x^4\sin x)\) 
solution 
Practice 30 

\(\displaystyle{y=[\log_4(1+e^x)]^2}\) 
solution 
Practice 31 

\(\displaystyle{p(x)=\ln\left(x^2\cdot\sqrt{x^3+3x}\cdot(x+2)^4\right)}\) 
solution 