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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

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

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

    \[\begin{align} (xe^x)'&=(x)'(e^x)+x(e^x)' \\ &= e^x+xe^x. \end{align}\]

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

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

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

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

    Larry Green (Lake Tahoe Community College)

    • Integrated by Justin Marshall.