6.3: Derivatives of Exponential Functions

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Nov 11, 2024
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- 6.2: Related Rates
- 7: Differentiation - Increasing and Decreasing Values and Extrema

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Exponential functions, and the rate of change, are used to model many real-world situations such as population growth, radioactive half-life decay, attenuation of electromagnetic signals in media, and financial transactions. Do you know how to write general exponential equations for the growth of a population that doubles every 5 years, and its rate of change?

Derivatives of Exponential Functions

An exponential function $f(x) \nonumber$ has the form:

$f(x) = b^x \nonumber$

where b is called the base and is a positive, real number.

The figure below shows a few exponential function graphs for 0<b≤10. It is very clear that the sign of the derivative of an exponential depends on the value of b. If 0<b<1, the value of the derivative of the function (slope of the tangent line) will be negative because the function is always decreasing as x increases. For b>1, the derivative of the function will always be positive because the function increases as x increases.

Screen Shot 2020-11-12 at 11.45.54 PM.png

CC BY-NC-SA

But, what is the derivative of an exponential function? We can take the following steps to find an expression for $\dfrac{d}{dx}[bx] \nonumber$ by using the definition of the derivative:

$\displaystyle \dfrac{d}{dx}[bx]= \lim_{h \to 0} \dfrac{b^{x+h}−b^x}{h} \nonumber$ .... Limit definition of the derivative

$= \lim_{h \to 0} \dfrac{b^xb^h−b^x}{h} \nonumber$ ....Exponent property

$= \lim_{h \to 0} \dfrac{b^h−1}{h}⋅b^x \nonumber$ ....Factoring

$= ( \lim_{h→0} \dfrac{b^h−1}{h})⋅bx \nonumber$ ....Limit of a product property

The result above shows $\dfrac{d}{dx}[bx] \nonumber$ depends on the product of $\lim_{h→0} \dfrac{b^h−1}{h} \nonumber$ and the original function. But what is $\lim_{h→0} \dfrac{b^h−1}{h} \nonumber$ ? There are a number of ways to evaluate this limit, but for now let’s take a quick look at the behavior of $\dfrac{b^h−1}{h} \nonumber$ . This function is graphed below for a number of values of b, and the limit at $h=0 \nonumber$ is indicated by the points A−E.

Screen Shot 2020-11-21 at 6.04.44 PM.png

CC BY-NC-SA

While it is not at all obvious: $\lim_{h→0} \dfrac{b^h−1}{h} = ln(b) \nonumber$ . Remember the natural logarithm function $y=lnx \nonumber$ on your calculator? The natural logarithm is the general logarithm function with base $b=e=2.71828 \nonumber$ ...

Given the exponential function $f(x)=b^x \nonumber$ , where the base b is a positive, real number, then the general representation of the derivative of an exponential function is:

$\dfrac{d}{dx}[bx]=lnb⋅b^x \nonumber$

Adding the Chain Rule to the definition, given the exponential function f(x)=bu, where u=g(x) and g(x) is a differentiable function, then:

$\dfrac{d}{dx}[bu]=(lnb⋅b^u) \dfrac{du}{dx} \nonumber$

Examples

Example 1

Earlier, you were asked what the general exponential equation for the growth of a population that doubles every 5 years is.

A population $P(t) \nonumber$ that doubles every 5 years could be modeled as $P(t)=P_02^{\dfrac{t}{5}} \nonumber$ , where the variable t represents number of years since the population was at a level of $P_0 \nonumber] \nonumber$ . Were you able to determine that the rate of change of $P(t) \nonumber$ is $P′(t) = \dfrac{P_0 ln2}{5}⋅2^{\dfrac{t}{5}} \nonumber$ ?

Example 2

Given $y=500⋅0.7^x \nonumber$ , what is $\dfrac{dy}{dx} \nonumber$ ?

$\dfrac{dy}{dx}=\dfrac{d}{dx}[500⋅0.7^x] \nonumber$

$= 500 \dfrac{d}{dx}[0.7^x] \nonumber$

$= 500[ln(0.7)⋅0.7^x] \nonumber$ ....Use your calculator to find ln(0.7)

$=−178.3⋅0.7^x \nonumber$

Hence, $\dfrac{dy}{dx} =−178.3⋅0.7^x \nonumber$ , and as expected, the slopes of all tangent lines are negative.

There is an important special case that you must know about

Example 3

Given $y=500⋅e^x \nonumber$ , what is $\dfrac{dy}{dx} \nonumber$ ?

$\dfrac{dy}{dx}= \dfrac{d}{dx}[500⋅ex] \nonumber$

$= 500 \dfrac{d}{dx}[ex] \nonumber$

$\dfrac{dy}{dx}=500[ln(e)⋅ex] \nonumber$ ....Use your calculator to find ln(e)

$= \dfrac{d}{dx}[500⋅ex] \nonumber$

$=500[1⋅ex] \nonumber$

$=500⋅ex \nonumber$

Hence, $\dfrac{dy}{dx}=500⋅ex \nonumber$ , and this is just the original function. This exponential function, with base e, is special: the rate of change (or slope of the tanget line) at any point is equal to the value of the function at that point.

Example 4

Given $y=10⋅2.5−3x^2 \nonumber$ , what is $\dfrac{dy}{dx} \nonumber$ ?

$\dfrac{dy}{dx} = \dfrac{d}{dx}[10⋅2.5−3x2] \nonumber$

$=10 \dfrac{d}{dx}[2.5−3x2] \nonumber$

$=10⋅ln(2.5)⋅2.5−3x2⋅ddx[−3x^2] \nonumber$

$=10⋅(0.9162)⋅2.5−3x^2⋅[−6x] \nonumber$

$=−55x⋅2.5−3x^2 \nonumber$

Therefore, $\dfrac{dy}{dx} =−55x⋅2.5−3x^2 \nonumber$

Example 5

Given $y=500⋅e−2x⋅cos(5πx) \nonumber$ , what is $\dfrac{dy}{dx} \nonumber$ ?

$\dfrac{dy}{dx}= \dfrac{d}{dx}[500⋅e−2x⋅cos(5πx)] \nonumber$

$=500⋅[ \dfrac{d}{dx}(e−2x)⋅cos(5πx)+e−2x⋅ \dfrac{d}{dx}(cos(5πx))] \nonumber$ ....Product Rule

$=500⋅[ln(e)⋅e−2x \dfrac{d}{dx}(−2x)⋅cos(5πx)+e−2x⋅(−sin(5πx)⋅ \dfrac{d}{dx}(5πx))] \nonumber$ ....Use Chain Rule

$=500⋅[(1)⋅e−2x⋅(−2)⋅cos(5πx)+e−2x⋅(−sin(5πx)⋅5π)] \nonumber$ ....Simplify.

$=−500⋅e−2x[2⋅cos(5πx)+5πsin(5πx)] \nonumber$ ...Simplify

Therefore, $\dfrac{dy}{dx} =−500⋅e−2x[2⋅cos(5πx)+5πsin(5πx)] \nonumber$ .

Review

For #1-14, find the derivative.

$y=7^x \nonumber$
$y=3^{2x} \nonumber$
$y=5^x−3x^2 \nonumber$
$y=2^{x^2} \nonumber$
$y=e^{x^2} \nonumber$
$f(x)= \dfrac{1}{ \sqrt{πσ}}e^{−αk(x−x0)^2} \nonumber$ where σ, α, x0, and k are constants and σ≠0.
$y=e^{6x} \nonumber$
$y=e^{3x^3}−2x^2+6 \nonumber$
$y=\dfrac{e^x−e−x}{e^x+e−x} \nonumber$
$y=cos(e^x) \nonumber$
$y=e^{−x}3^x \nonumber$
$y=3^{−x^2+2x+1} \nonumber$
$y=2^x3^x \nonumber$
$y=e^{−x}sinx \nonumber$
Find an equation of the tangent line to $f(x)=x^3+2e^x \nonumber$ at the point (0, 2).

Review (Answers)

To see the Review answers, open this PDF file and look for section 3.9.

Vocabulary

Term	Definition
derivative	The derivative of a function is the slope of the line tangent to the function at a given point on the graph. Notations for derivative include f′(x), dydx, y′, dfdx and \dfrac{df(x)}{dx}.
Exponential Function	An exponential function is a function whose variable is in the exponent. The general form is $y=a⋅b^{x−h}+k \nonumber$ .

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