Derivatives of Exponentials

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Lesson Priority: VIP Knowledge

 
Objectives
  • Learn how to take the derivative of $e^x$
  • Modify these new facts to take the derivative of things other than base $e$
  • Know the rule for the derivative of $e^{kx}$ where $k$ is a constant
Lesson Description

Here we will learn how to take the derivative of exponential terms, starting with the foundational natural exponentiation function $e^x$. After studying the derivative pattern of this vital function, we will see how to take the derivative of similar functions, including exponentials with bases other than $e$ as well as exponentials with constant multipliers on the variable, such as $e^{5x}$.

 
Practice Problems

Practice problems and worksheet coming soon!

 

Exponentials - The Most Meta

Loosely speaking, exponential functions are the way they are because each function value in an exponential corresponds closely with the function's instantaneous slope at that same point. In this way, it's fair to say that exponential functions are very meta, and it is interesting (to a mathematician anyway) that this family of functions is the only family that has this property. In a way, one can say that exponential functions are the result of constructing functions whose values are closely related to (or in some cases exactly equal to) the instantaneous slope at the same point, rather than consider the characteristic of these functions to be a consequence.First, let's understand the simplest case, which happens to be an exponential function with base $e$.
Define: Natural Exponential DerivativeLet $f(x) = e^x$. Then,$$f'(x) = e^x$$
In other words, the derivative of $e^x$ is $e^x$. This particular function has the property that its derivative is itself. We won't rigorously prove it here and now, but it turns out that $f(x) = e^x$ is the ONLY function whose derivative is exactly itself (as well as any constant multiple of $e^x$, such as $2e^x$).Very quickly, you're going to be tempted to nominate exponential functions with the award for easiest function to differentiate. There's certainly an argument to be made there, but we will continue to deal with and use many of the same rules and concepts as we've seen with other derivatives up to this point - so my advice coming right out of the gate is, don't get overconfident!
Did You Know?
Many mathematicians refer to $e$ as "the natural number". This may seem like a strange choice of words, but one of the aspects of this number that make it so special is that it is the only number that when used as the base of an exponential function yields a function whose function values exactly match its instantaneous slope. In other words, the only function in existence whose derivative is itself is$$f(x) = e^x$$

Derivatives of Any-Base Exponentials

The derivative of a general exponential term of the form $b^x$ (where $b$ is any real number greater than $0$, but not $1$) has a similar form to the derivative of $e^x$.
Define: General Exponential Derivatives$$\frac{d}{dx} \,\, b^x = b^x \cdot \ln(b)$$
In fact, if we replace $b$ with $e$ in that definition, we get$$\frac{d}{dx} \,\, e^x = e^x \cdot \ln(e)$$$$=e^x \cdot 1 = e^x$$Of course, this aligns with what we saw earlier about the derivative of $e^x$.Here are some examples.Examples 1-4Calculate each derivative.
 
Example 1$$\frac{d}{dx} \,\, 4^x$$$\blacktriangleright$ This is a straight-forward application of the derivative definition above.$$\Rightarrow 4^x \cdot \ln(4)$$
 
Example 2$$\frac{d}{dx} \,\, -5e^x$$$\blacktriangleright$ By the constant rule of derivatives, the $-5$ coefficient stays along for the ride, and we just have to calculate the derivative of the rest of the expression ($e^x$, whose derivative is $e^x$).$$\Rightarrow -5e^x$$
 
Example 3$$\frac{d}{dx} \,\, -3 \left( \pi^x \right)$$$\blacktriangleright$ This problem also requires us to proceed using the constant rule of derivatives. Additionally, instead of the natural number base, we have a base of $\pi$. Therefore:$$\frac{d}{dx} \,\, \pi^x = \pi^x \cdot \ln(\pi)$$$$\therefore \frac{d}{dx} \,\, -3 \left( \pi^x \right) = -3 \left( \pi^x \right) \cdot \ln(\pi)$$
 
Example 4$$\frac{d}{dx} \,\, 3e^x - 100^x + 4x^3$$$\blacktriangleright$ This problem is simply a combination of two exponential derivatives, and one polynomial term derivative which is computed using our recently gleaned knowledge of the Power Rule », as well as the sum and constant derivative rules.$$\Rightarrow 3e^x - 100^x \cdot \ln(100) + 12x^2$$

Derivatives of General Exponentials

Typically, when we learn about exponentials as a function family, we study functions of the form$$f(x) = a \left( b^{kx} \right) + c$$If we are studying derivative of $f(x)$, however, the $+c$ term is useless, since the derivative of a lone constant is zero. We should therefore turn our attention to being able to take derivatives of the form $a \left( b^{kx} \right)$.
Define: Exponents of the Form "kx"Let $k$ be any real number constant, and let $b$ be any positive real number except $1$.$$\frac{d}{dx} \,\, e^{kx} = ke^{kx}$$$$\frac{d}{dx} \,\, b^{kx} = kb^{kx}\cdot \ln(b)$$It follows that$$\frac{d}{dx} \,\, ae^{kx} = ake^{kx}$$$$\frac{d}{dx} \,\, ab^{kx} = akb^{kx}\cdot \ln(b)$$
By the constant rule of derivatives », the $a$ coefficient stays, so that part of the result shouldn't be surprising. Obtaining the rest of the results will be more clear when we learn the Chain Rule ». For now we will accept them at face value.Examples 5-8Calculate each derivative.
 
Example 5$$\frac{d}{dx} \,\, e^{3x}$$$\blacktriangleright$ This is a straight-forward application of the definition above.$$\frac{d}{dx} \,\, e^{3x} = 3e^{3x}$$
 
Example 6$$\frac{d}{dx} \,\, 6^{7x}$$$\blacktriangleright$ Once again, this is a straight-forward application of the definition.$$\frac{d}{dx} \,\, 6^{7x} = 7 \left(6^{7x} \right) \cdot \ln(6)$$
 
Example 7$$\frac{d}{dx} \,\, 7e^{4x}$$$\blacktriangleright$ The $7$ constant will stay with the expression, and the $4$ in the exponent is going to come down as a multiplier, per the rule. Altogether, we have$$\frac{d}{dx} \,\, 7e^{4x} = 28e^{4x}$$
 
Example 8$$\frac{d}{dx} \,\, 2\left(5^{4x}\right)$$$\blacktriangleright$ Similar to example 7, we will keep the outside coefficient along for the ride, and focus on the derivative of $5^{4x}$.$$\Rightarrow 2\cdot 4\cdot \left(5^{4x}\right) \cdot \ln(5) = 8 \left(5^{4x}\right) \cdot \ln(5)$$

Put It To The Test

Try each of the following problems yourself before checking the solutions.
 
Example 12$$\frac{d}{dx} \,\, e^{\pi^2 x}$$
Show solution
$\blacktriangleright$ As we've learned, to calculate the derivative of $e^{kx}$, we need to bring the constant down in front. Just because the constant is $\pi^2$ doesn't mean it's any different!$$\frac{d}{dx} \,\, e^{\pi^2 x} = \pi^2 e^{\pi^2 x}$$
 
Example 13$$\frac{d}{dx} \,\, 3e^{-3x}$$
Show solution
$\blacktriangleright$ This is a direct application of preceding definitions. Don't be careless with the negative sign, since the constant in the exponent is $-3$, not just $3$.$$\frac{d}{dx} \,\, 3e^{-3x} = -3 \cdot 3e^{-3x}$$$$=-9e^{-3x}$$
 
Example 14$$\frac{d}{dx} \,\, -\left(2^{x+1}\right)$$
Show solution
$\blacktriangleright$ While we didn't yet learn how to take the derivative of exponentials with exponents modified to include sum expressions, this one can be handled by rearranging things with exponent laws.$$2^{x+1} = 2^x \cdot 2^1$$$$\therefore \frac{d}{dx} \,\, -\left(2^{x+1}\right) = \frac{d}{dx} -2 \left(2^{x}\right) = -2\left(2^{x}\right) \cdot \ln(2)$$
 
Example 15$$\frac{d}{dx} \,\, 5 \left(5^{5x}\right)$$
Show solution
$\blacktriangleright$ There are two major details to keep track of here. First, remembering how to deal with the exponent that has a constant coefficient with the $x$, and second, the fact that this exponential is not base $e$.$$\frac{d}{dx} \,\, 5 \left(5^{5x}\right) = 25\left( 5^{5x}\right) \ln(5)$$
 
Example 16$$\frac{d}{dx} \,\, 4^{ex}$$
Show solution
$\blacktriangleright$ The key to this problem is to remember that $e$ is just a number, even thought it is a special one.$$\frac{d}{dx} \,\, 4^{ex} = e \left(4^{ex}\right)\cdot \ln(4)$$
 
Example 17$$\frac{d}{dx} \,\, -6 \ln(x)$$
Show solution
$$\blacktriangleright \,\, \frac{-6}{x}$$
 
Example 18$$\frac{d}{dx} \,\, 10 \log(x)$$
Show solution
$\blacktriangleright$ Don't forget that when a logarithm is written without the base, we are supposed to assume it is base $10$. Therefore, following the defintion of derivatives for logarithms that are not base $e$, we have$$\frac{d}{dx} \,\, 10 \log(x) = \frac{10}{x \ln(10)}$$
 
Example 19$$\frac{d}{dx} \,\, \log_4 (18x)$$
Show solution
$\blacktriangleright$ While we didn't address derivatives of the form $\log_b (cx)$ in the definitions and examples of this lesson, and while we will learn the fast-track way to deal with these derivatives in the Chain Rule » lesson, there is a fairly simple way to get to the answer here, using logarithm rules.$$\log_4 (18x) = \log_4(18) + \log_4(x)$$Therefore,$$\frac{d}{dx} \,\, \log_4 (18x) = \frac{d}{dx} \,\, \log_4(18) + \frac{d}{dx} \,\, \log_4(x)$$And since $\log_4(18)$ is a constant, its derivative is zero:$$\frac{d}{dx} \,\, \log_4 (18x) = \cancel{\frac{d}{dx} \,\, \log_4(18)} + \frac{d}{dx} \,\, \log_4(x)$$$$=\frac{1}{x\ln(4)}$$
 
Lesson Takeaways
  • Know how to take the derivative of any natural exponential expression, with and without constants, with $x$ or $kx$ in the exponent
  • Know how to modify the case for natural exponentials to be able to take the derivative of general base exponentials
  • Take the derivative of single, simple logarithm expressions, and be able to generalize the simpler case for natural logs to the general case for a log with any constant base

Lesson Metrics

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Headlines - Every lesson is subdivided into mini sections that help you go from "no clue" to "pro, dude". If you keep getting lost skimming the lesson, start from scratch, read through, and you'll be set straight super fast.

Examples - there is no better way to learn than by doing. Some examples are instructional, while others are elective (such examples have their solutions hidden).

Perils and Pitfalls - common mistakes to avoid.

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Put It To The Test - Shows you examples of the most common ways in which the concept is tested. Just knowing the concept is a great first step, but understanding the variation in how a concept can be tested will help you maximize your grades!

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