17Calculus Derivatives - Exponential Functions

17Calculus

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.

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.

3Blue1Brown - Derivatives of exponentials [13min-49secs]

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.

PatrickJMT - Derivatives of Exponential Functions [5min-49secs]

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.

Dr Chris Tisdell - Derivative of exponentials [9min-15secs]

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.

MathTV - Some Natural Exponential Functions and Tangent Lines [4min-11secs]

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

Krista King Math - 1069 video 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

PatrickJMT - 1074 video 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

PatrickJMT - 1076 video solution

video by PatrickJMT

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Practice - Chain Rule Required

$$f(x) = (x^2-1)e^{-x}$$

Problem Statement

Calculate the derivative of $$f(x) = (x^2-1)e^{-x}$$

Solution

Krista King Math - 1066 video 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

Krista King Math - 1067 video solution

video by Krista King Math

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$$\displaystyle{ f(x) = \frac{1-e^{-x}}{x} }$$

Problem Statement

Calculate the derivative of $$\displaystyle{ f(x) = \frac{1-e^{-x}}{x} }$$

Solution

Krista King Math - 1068 video 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. $$\displaystyle{ 3e^{ x^2+7 } }$$

$$\displaystyle{ \frac{d}{dx} \left[ 3e^{ x^2+7 } \right] = 6xe^{x^2+7} }$$

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 $$u=x^2+7$$ $$\displaystyle{ \frac{d}{dx} \left[ 3e^{ x^2+7 } \right] }$$ $$\displaystyle{ 3\frac{d[e^u]}{du} \cdot \frac{d[x^2+7]}{dx} }$$ $$3e^u \cdot (2x)$$ $$6xe^{x^2+7}$$

$$\displaystyle{ \frac{d}{dx} \left[ 3e^{ x^2+7 } \right] = 6xe^{x^2+7} }$$

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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

PatrickJMT - 1077 video 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

PatrickJMT - 1075 video 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

PatrickJMT - 1078 video 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

PatrickJMT - 1079 video solution

video by PatrickJMT

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$$\displaystyle{ y = \cos \left( \frac{1-e^{2x}}{1+e^{2x}} \right) }$$

Problem Statement

Calculate the derivative of this function and give your final answer in completely factored form. $$\displaystyle{ y = \cos \left( \frac{1-e^{2x}}{1+e^{2x}} \right) }$$

Solution

Krista King Math - 986 video 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 $$\dfrac{d}{dx}( 2^{x^2} )$$ .

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

Problem Statement

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

Hint

$$2 = e^{\ln2}$$

Solution

Michel vanBiezen - 4379 video solution

video by Michel vanBiezen

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

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