Derivatives of Exponential Functions
On this page we discuss the derivatives of the exponential functions \(e^x\) and \(a^x\). For the basic exponential derivatives you do not need the chain rule. But we include discussion that requires the chain rule on this page. However, 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.
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Topics You Need To Understand For This Page 

Related Topics and Links 
WikiBooks  Derivatives of Exponential and Logarithm Functions 
The Exponential Rule With Base \(e\)
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.
Before we go on, let's watch a video that gives an intuitive explanation of the derivative of exponential functions and why \(f(x)=e^x\) is special.
video by 3Blue1Brown 

The Exponential Rule With The General Base \(a\)
Okay, so what do you do 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 \(e\).
\( \begin{array}{rcl}
y & = & a^x \\
\ln(y) & = & \ln(a^x) \\
& = & x\ln(a) \\
e^{\ln(y)} & = & e^{x\ln(a)} \\
y & = & e^{x\ln(a)}
\end{array} \)
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.
Here is a video that goes through this explanation again. It is always good to get explanations from different sources since it will help you understand the material better.
video by PatrickJMT 

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.
video by Dr Chris Tisdell 

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.
video by MathTV 

Practice  Chain Rule Not Required
Unless otherwise instructed, calculate the derivative of these functions.
\( y = e^x(x+x\sqrt{x}) \)
Problem Statement
Calculate the derivative of \( y = e^x(x+x\sqrt{x}) \)
Solution
video by Krista King Math 

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\( f(x) = 4^x+3e^x+x^4 \)
Problem Statement
Calculate the derivative of \( f(x) = 4^x+3e^x+x^4 \)
Solution
video by PatrickJMT 

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\( f(x) = e^x x^2 \)
Problem Statement
Calculate the derivative of \( f(x) = e^x x^2 \)
Solution
video by PatrickJMT 

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Practice  Chain Rule Required
\( f(x) = (x^21)e^{x} \)
Problem Statement
Calculate the derivative of \( f(x) = (x^21)e^{x} \)
Solution
video by Krista King Math 

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\(\displaystyle{ f(x)=xe^{\sqrt{x}} }\)
Problem Statement
Calculate the derivative of \(\displaystyle{ f(x)=xe^{\sqrt{x}} }\)
Solution
video by Krista King Math 

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\(\displaystyle{ f(x) = \frac{1e^{x}}{x} }\)
Problem Statement
Calculate the derivative of \(\displaystyle{ f(x) = \frac{1e^{x}}{x} }\)
Solution
video by Krista King Math 

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\(\displaystyle{ 3e^{ x^2+7 } }\)
Problem Statement 

Calculate the derivative of this function and give your final answer in completely factored form. \(\displaystyle{ 3e^{ x^2+7 } }\)
Final Answer 

\(\displaystyle{ \frac{d}{dx} \left[ 3e^{ x^2+7 } \right] = 6xe^{x^2+7} }\)
Problem Statement
Calculate the derivative of this function and give your final answer in completely factored form. \(\displaystyle{ 3e^{ x^2+7 } }\)
Solution
\(\displaystyle{ \frac{d}{dx} \left[ 3e^{ x^2+7 } \right] }\) 
\(\displaystyle{ 3e^{x^2+7} \cdot \frac{d}{dx}[x^2+7] }\) 
\( 3e^{x^2+7} \cdot (2x) \) 
\( 6xe^{x^2+7} \) 
Another way to work this is with the substitution method.
let \(u=x^2+7\) 
\(\displaystyle{ \frac{d}{dx} \left[ 3e^{ x^2+7 } \right] }\) 
\(\displaystyle{ 3\frac{d[e^u]}{du} \cdot \frac{d[x^2+7]}{dx} }\) 
\( 3e^u \cdot (2x) \) 
\( 6xe^{x^2+7} \) 
Final Answer
\(\displaystyle{ \frac{d}{dx} \left[ 3e^{ x^2+7 } \right] = 6xe^{x^2+7} }\)
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\(\displaystyle{ f(x) = e^{x\sin(2x)} }\)
Problem Statement
Calculate the derivative of \(\displaystyle{ f(x) = e^{x\sin(2x)} }\)
Solution
video by PatrickJMT 

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\(\displaystyle{ g(x) = 2e^{\cos(x)\sin(5x)} }\)
Problem Statement
Calculate the derivative of \(\displaystyle{ g(x) = 2e^{\cos(x)\sin(5x)} }\)
Solution
video by PatrickJMT 

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\(\displaystyle{f(t)=\cos\left(2^{\pi t}\right) }\)
Problem Statement
Calculate the derivative of \(\displaystyle{f(t)=\cos\left(2^{\pi t}\right) }\)
Solution
video by PatrickJMT 

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For what values of \( x \) does \( h(x)=5e^{5x}25x \) have negative derivatives?
Problem Statement
For what values of \( x \) does \( h(x)=5e^{5x}25x \) have negative derivatives?
Solution
video by PatrickJMT 

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\(\displaystyle{ y = \cos \left( \frac{1e^{2x}}{1+e^{2x}} \right) }\)
Problem Statement
Calculate the derivative of this function and give your final answer in completely factored form. \(\displaystyle{ y = \cos \left( \frac{1e^{2x}}{1+e^{2x}} \right) }\)
Solution
video by Krista King Math 

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\( \dfrac{d}{dx}( 2^{x^2} ) \)
Problem Statement 

Evaluate \( \dfrac{d}{dx}( 2^{x^2} ) \) .
Hint 

\( 2 = e^{\ln2} \)
Problem Statement 

Evaluate \( \dfrac{d}{dx}( 2^{x^2} ) \) .
Final Answer 

\( \dfrac{d}{dx}( 2^{x^2} ) \) \( = 2^{x^2} 2x\ln2 = 2^{x^2+1}x\ln2 \)
Problem Statement
Evaluate \( \dfrac{d}{dx}( 2^{x^2} ) \) .
Hint
\( 2 = e^{\ln2} \)
Solution
video by Michel vanBiezen 

Final Answer
\( \dfrac{d}{dx}( 2^{x^2} ) \) \( = 2^{x^2} 2x\ln2 = 2^{x^2+1}x\ln2 \)
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