Skip to main content
Mathematics LibreTexts

11.4: Derivatives

  • Page ID
    90287
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    Now that we know some elementary functions, we seek their derivatives. We will not spend time exploring the appropriate limits in any rigorous way. We are only interested in the results. We provide these in Table \(\PageIndex{1}\). We expect that you know the meaning of the derivative and all of the usual rules, such as the product and quotient rules.

    Table \(\PageIndex{1}\): Table of Common Derivatives (\(a\) is a constant).

    Function Derivative
    \(a\) 0
    \(x^{n}\) \(n x^{n-1}\)
    \(e^{a x}\) \(a e^{a x}\)
    \(\ln a x\) \(\frac{1}{x}\)
    \(\sin a x\) \(a \cos a x\)
    \(\cos a x\) \(-a \sin a x\)
    \(\tan a x\) \(a \sec ^{2} a x\)
    \(\csc a x\) \(-a \csc a x \cot a x\)
    \(\sec a x\) \(a \sec a x \tan a x\)
    \(\cot a x\) \(-a \csc ^{2} a x\)
    \(\sinh a x\) \(a \cosh a x\)
    \(\cosh a x\) \(a \sinh a x\)
    \(\tanh a x\) \(a \operatorname{sech}^{2} a x\)
    \(\operatorname{csch} a x\) \(-a \operatorname{csch} a x \operatorname{coth} a x\)
    \(\operatorname{sech} a x\) \(-a \operatorname{sech} a x \tanh a x\)
    \(\operatorname{coth} a x\) \(-a \operatorname{csch}^{2} a x\)

    Also, you should be familiar with the Chain Rule. Recall that this rule tells us that if we have a composition of functions, such as the elementary functions above, then we can compute the derivative of the composite function. Namely, if \(h(x)=f(g(x))\), then \[\frac{d h}{d x}=\frac{d}{d x}(f(g(x)))=\left.\frac{d f}{d g}\right|_{g(x)} \frac{d g}{d x}=f^{\prime}(g(x)) g^{\prime}(x) .\label{eq:1}\]

    Example \(\PageIndex{1}\)

    Differentiate \(H(x)=5 \cos \left(\pi \tanh 2 x^{2}\right)\).

    Solution

    This is a composition of three functions, \(H(x)=f(g(h(x)))\), where \(f(x)=\) \(5 \cos x, g(x)=\pi \tanh x\), and \(h(x)=2 x^{2}\). Then the derivative becomes \[\begin{align} H^{\prime}(x)&=5\left(-\sin \left(\pi \tanh 2 x^{2}\right)\right) \frac{d}{d x}\left(\left(\pi \tanh 2 x^{2}\right)\right)\nonumber \\  &=-5 \pi \sin \left(\pi \tanh 2 x^{2}\right) \operatorname{sech}^{2} 2 x^{2} \frac{d}{d x}\left(2 x^{2}\right)\nonumber \\ &=-20 \pi x \sin \left(\pi \tanh 2 x^{2}\right) \operatorname{sech}^{2} 2 x^{2} .\label{eq:2} \end{align}\]


    This page titled 11.4: Derivatives is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by Russell Herman via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

    • Was this article helpful?