Differentiation and Integration of the Natural Exponential Function

The next presentation illustrates several exponential graphs and serves as a reminder of the importance of applications involving exponential functions in real-world contexts.

Natural Exponential Function and its Graph

The natural exponential function is a function of the form f(x) = e^x for every number x. Recall the number e is an irrational number

Did You Know?
Euler, the famous 19th century Swiss mathematician called the numbers e and pi transcendental numbers because "they transcended the power of algebraic methods" e is often called Euler's number

with decimal approximation beginning 2.71828182846... The abbreviation exp x is sometimes used to represent f(x) = e^x. The natural exponential function f(x) = e^x is a special case of the general exponential functions equation image indicator with a = e > 1.

Properties of the Natural Exponential Function

The graph of y = e^x confirms that e^x possesses the same characteristics as general exponential functions with base a > 1. The natural exponential function f(x) = e^x is an increasing continuous function with domain equation image indicator $LaTeX: \left(-\infty,\infty\right)$ and range $LaTeX: \left(0,\infty\right)$ , and it passes through the point (0, 1). Recall that a function f is described as increasing on an interval I if for any two numbers and in the interval, $LaTeX: x_1<x_2\rightarrow f\left(x_1\right)<f\left(x_2\right)$ . As with general exponential functions with base a > 1, equation image indicator $LaTeX: \lim_{x\to\infty }e^x=0 \:and\:\lim_{x\to\infty }e^x =\infty$ .

graph of e^x

Derivative of the Natural Exponential Function

A unique characteristic of the natural exponential function is that its instantaneous rate of change is proportional to itself. Many real-life varying quantities such as undisturbed bank balances and populations exhibit this same characteristic. Stated another way, the natural exponential function is its own derivative. This is expressed as equation image indicator $LaTeX: \frac{d}{dx}\left(e^x\right)=e^x$ . More generally, if u is a differentiable function of x, then $LaTeX: \frac{d}{dx}\left(e^u\right)=e^u\frac{du}{dx}$ . Use the interactive applet to explore the relationship between an exponential function and its derivative by CLICKING HERE.

A proof of the derivative of the natural exponential function follows.

View a presentation of a slightly different approach to proving the derivative of the natural exponential function along with application problems and calculator-based computation of the numerical derivative.

The slope of a tangent line to an implicitly defined transcendental function is often needed, which necessitates the application of the technique of implicit differentiation illustrated in the following video.

A presentation on the application of the Chain Rule when differentiating the transcendental functions e^x and ln x follows.

Integration of the Natural Exponential Function

Because the derivative of a natural exponential function is itself, equation image indicator $LaTeX: \frac{d}{dx}\left(e^x\right)=e^x$ , its integral is $LaTeX: \int e^xdx=e^x+C$ . In general, if u is a differentiable function of x, $LaTeX: \int e^udx=e^u+C$ . Examples of integration involving natural exponential functions and area under a natural exponential function curve are presented below.

Example 1

text annotation indicator $LaTeX: \int_0^1e^xdx$

Solution: $LaTeX: ]int_0^1e^xdx=ex\Big]_0^1=e^1-e^0=e-1$

Example 2

text annotation indicator $LaTeX: \int\frac{e^{\frac{1}{x}}}{x^2}dx$

Solution: Let $LaTeX: y=\frac{1}{x};du=\left(-\frac{1}{x^x}\right)dx$

$LaTeX: \int \frac{e^{\frac{1}{x}}}{x^2}dx=-\int e^udu=-e^u+C=-e^{\frac{1}{x}}+C$

Example 3

text annotation indicator $LaTeX: \int\sec^2\theta c^{\tan\theta}d\theta$

Solution: Let $LaTeX: u=\tan\theta;\:du=\sec^2\theta d\theta$

$LaTeX: \int\sec^2\theta e^{\tan\theta}d\theta=e^{\tan\theta}+C$

Example 4

Find the area bounded by the curve text annotation indicator , y = 0, x = 0, and x = 1.

Solution:

$LaTeX: Let\:u=-x^2;\:du=-2xdx;\:xdx=\frac{du}{-2}$

$LaTeX: \int_0^1xe^{-x^2}dx=-\frac{1}{2}e^{-x^2}\Big]_0^1=-\frac{1}{2}\left(e^{-1}-1\right)=-\frac{1}{2}\left(\frac{1}{e}-1\right)=-\frac{1}{2e}+\frac{1}{2}$

Example 5

text annotation indicator $LaTeX: \int\frac{e^x-5}{e^x}dx$

Solution: Let u = -x, du = -dx

$LaTeX: \int\frac{e^x-5}{e^x}dx=\int\left(1-5\int e^{-x}\right)dx=dx+5e^{-x}+C=x+\frac{5}{e^x}+C$

A review of integration using substitution involving algebraic, natural logarithmic, and natural exponential functions follows.

Differentiation and Integration of the Natural Exponential Function Practice

Differentiation and Integration of the Natural Exponential Function: Even More Problems!

Complete problems from your textbook and/or online resources as needed to ensure your complete understanding of differentiation and integration of the natural exponential function.

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