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Mathematics LibreTexts

4.5: The Derivative and Integral of the Exponential Function

Definitions and Properties of the Exponential Function

The exponential function, 

\[y=e^x \]

is defined as the inverse of

\[\ln x.\]

Therefore

\[\ln(e^x) = x \]

and 

\[e^{\ln x} =x. \]

Recall that

  1. \[e^ae^b=e^{a+b}\]

  2. \[\dfrac{e^a}{e^b}=e^{(a-b)}.\]

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

\[g'(x)=\dfrac{1}{f'(g(x))}. \]

Let 

\[f(x)=\ln(x) \]

then

\[f'(x)=\dfrac{1}{x} \]

so that

\[f'(g(x))=\dfrac{1}{e^x}. \]

Hence

\[g'(x)=e^x \]

Theorem:

If

\[f(x)=e^x  \]

then

\[f'(x)=f(x)=e^x\]

Example 1

Find the derivative of

\[e^{2x}.\]

Solution

We use the chain rule with

\[y = e^u,  \;\;  u = 2x.\]

Which gives

\[y'=e^u, \;\; u'=2.\]

So that

\[(e^{2x})'=(e^u)(2)=2e^{2x}.\]

Example 2

Find the derivative of \[xe^x.\]

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 

  1. \[ln(e^x) \]

  2. \[\dfrac{e^x}{x^2}.\]

Example 3

\[\int e^x \; dx \]

Solution

Since

\[e^x = (e^x)' \]

We can integrate both sides to get

\[\int e^x \; dx = e^x +C \]

Example 4

\[\int e^xe^{e^x}\; dx \]

Solution

For this integral, we can use \(u\) substitution with

\[u=e^x, \;\; du=e^x \; dx. \]

The integrals becomes

\[\begin{align} \int e^u \; du &= e^u +C \\ &= e^{e^x}+C.  \end{align}\]

Exercise

Integrate:

  1. \[\int xe^{x^2} \; dx \]

  2. \[\int \dfrac{e^x}{1-e^x} \; dx. \]

Contributors

  • Integrated by Justin Marshall.