4.5: The Derivative and Integral of the Exponential Function
- Page ID
- 533
Definitions and Properties of the Exponential Function
The exponential function,
\[y=e^x \nonumber \]
is defined as the inverse of
\[\ln x.\nonumber \]
Therefore
\[\ln(e^x) = x \nonumber \]
and
\[e^{\ln x} =x. \nonumber \]
Recall that
-
\[e^ae^b=e^{a+b}\nonumber \]
-
\[\dfrac{e^a}{e^b}=e^{(a-b)}.\nonumber \]
Proof of 2:
\[\begin{align*} \ln\Big[\dfrac{e^a}{e^b}\Big] &= \ln[e^a]-\ln[e^b] \\ &= a-b \\ &= \ln[e^{a-b}] \end{align*}\]
since \(\ln(x)\) is 1-1, the property is proven.
The Derivative of the Exponential
We will use the derivative of the inverse theorem to find the derivative of the exponential. The derivative of the inverse theorem says that if \(f\) and \(g\) are inverses, then
\[g'(x)=\dfrac{1}{f'(g(x))}. \nonumber \]
Let
\[f(x)=\ln(x) \nonumber \]
then
\[f'(x)=\dfrac{1}{x} \nonumber \]
so that
\[f'(g(x))=\dfrac{1}{e^x}. \nonumber \]
Hence
\[g'(x)=e^x \nonumber \]
Theorem
If
\[f(x)=e^x \nonumber \]
then
\[f'(x)=f(x)=e^x\nonumber \]
Example 1
Find the derivative of
\[e^{2x}.\nonumber \]
Solution
We use the chain rule with
\[y = e^u, \;\; u = 2x.\nonumber \]
Which gives
\[y'=e^u, \;\; u'=2.\nonumber \]
So that
\[(e^{2x})'=(e^u)(2)=2e^{2x}.\nonumber \]
Example 2
Find the derivative of \[xe^x.\nonumber \]
Solution
We use the product rule:
\[\begin{align*} (xe^x)'&=(x)'(e^x)+x(e^x)' \\ &= e^x+xe^x. \end{align*} \]
Exercise
Find the derivatives of
-
\[ln(e^x) \nonumber \]
-
\[\dfrac{e^x}{x^2}.\nonumber \]
Example 3
\[\int e^x \; dx \nonumber \]
Solution
Since
\[e^x = (e^x)' \nonumber \]
We can integrate both sides to get
\[\int e^x \; dx = e^x +C \nonumber \]
Example 4
\[\int e^xe^{e^x}\; dx \nonumber \]
Solution
For this integral, we can use \(u\) substitution with
\[u=e^x, \;\; du=e^x \; dx. \nonumber \]
The integrals becomes
\[\begin{align*} \int e^u \; du &= e^u +C \\ &= e^{e^x}+C. \end{align*}\]
Exercise
Integrate:
- \(\int xe^{x^2} \; dx \nonumber \)
- \(\int \dfrac{e^x}{1-e^x} \; dx. \nonumber \)
Contributors and Attributions
- Larry Green (Lake Tahoe Community College)
Integrated by Justin Marshall.