Kevin Gonzales, Eric Hopkins, Catherine Zimmitti, Cheryl Kane, Modified to fit Applied Calculus from Coordinated Calculus by Nathan Wakefield et. al., Based upon Active Calculus by Matthew Boelkins

Section3.1Exponential Functions

Motivating Questions

What is an exponential function? What important properties does the graph have?

How do we take the derivative of an exponential function, and more complicated functions involving sums, products, quotients, and composition with exponential functions?

In this chapter we define exponential functions and their derivatives.

SubsectionExponential Functions

We will start with a formal definition of the exponential function.

Exponential Function

\begin{equation*}
f(x)= a \cdot e^x \text{, where }
e \approx 2.718281828459... \text{ (an irrational number) and } a \ne 0.
\end{equation*}

The constant \(a \) is the \(y\)-value of the \(y\)-intercept of the function.

To understand the exponential function it is good to examine the graph of \(f(x)=e^x\text{.}\)

Some important things to note about the function \(f(x)=e^x\text{:}\)

Find the derivative of the function \(f(x)=e^{x^3}\text{.}\)

To find the derivative of \(f(x)\) will use the chain rule where the outside function is \(e^x\) and the inside function is \(x^3\text{.}\) Since \(e^x\) is its own derivative we have:

Since \(f(x)=ae^x\) is its own derivative, the original term remains and we multiply by the derivative of the exponent.

Example3.4

Differentiate each of the following functions. State the rule(s) you use, label relevant derivatives appropriately, and be sure to clearly identify your overall answer.

The y-value the for the tangent line is \(f(1)=4e^{(1^2)}=4e\text{,}\) thus the tangent line is given by:

\begin{equation*}
y=8e(x-1)+4e
\end{equation*}

SubsectionSummary

An exponential function has the form \(f(x) = ae^x\text{.}\) The exponential function \(f(x) = e^x\) has some important properties: \(f(0) = 1 \text{,}\) \(\displaystyle \lim_{x \to -\infty} e^x = 0 \text{,}\) and \(\displaystyle \lim_{x \to \infty} e^x = \infty \text{.}\)

The derivative of \(f(x) = ae^x\) is \(f'(x) = ae^x\text{.}\) Combining this with the chain rule, we also see that the derivative of \(f(x) = ae^{g(x)} \) is \(f'(x) = ae^{g(x)}g'(x) \text{.}\)