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Mathematics LibreTexts

4.5: The Derivative and Integral of the Exponential Function

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    533
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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.