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# 4.5: The Derivative and Integral of the Exponential Function

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### 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. $\int \dfrac{e^x}{1-e^x} \; dx.$

Larry Green (Lake Tahoe Community College)

• Integrated by Justin Marshall.