8.4: Derivatives
- Page ID
- 91101
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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.
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
\[\dfrac{d h}{d x}=\dfrac{d}{d x}(f(g(x)))=\left.\dfrac{d f}{d g}\right|_{g(x)} \dfrac{d g}{d x}=f^{\prime}(g(x)) g^{\prime}(x) \nonumber \]
| Function | Derivative |
|---|---|
| \(a\) | 0 |
| \(x^{n}\) | \(n x^{n-1}\) |
| \(e^{a x}\) | \(a e^{a x}\) |
| \(\ln a x\) | \(\dfrac{1}{x}\) |
| \(\sin a x\) | \(a \cos a x\) |
| \(\cos a x\) | \(-a \sin a x\) |
| \(\tan a x\) | \(a \sec ^{2} a x\) |
| \(\csc a x\) | \(-a \csc a x \cot a x\) |
| \(\sec a x\) | \(a \sec a x \tan a x\) |
| \(\cot a x\) | \(-a \csc ^{2} a x\) |
| \(\sinh a x\) | \(a \cosh a x\) |
| \(\cosh a x\) | \(a \sinh a x\) |
| \(\tanh a x\) | \(a \operatorname{sech}^{2} a x\) |
| \(\operatorname{csch} a x\) | \(-a \operatorname{csch} a x \operatorname{coth} a x\) |
| \(\operatorname{sech} a x\) | \(-a \operatorname{sech} a x \tanh a x\) |
| \(\operatorname{coth} a x\) | \(-a \operatorname{csch}^{2} a x\) |
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
\[ \begin{aligned} H^{\prime}(x) &=5\left(-\sin \left(\pi \tanh 2 x^{2}\right)\right) \dfrac{d}{d x}\left(\left(\pi \tanh 2 x^{2}\right)\right) \\[4pt] &=-5 \pi \sin \left(\pi \tanh 2 x^{2}\right) \operatorname{sech}^{2} 2 x^{2} \dfrac{d}{d x}\left(2 x^{2}\right) \\[4pt] &=-20 \pi x \sin \left(\pi \tanh 2 x^{2}\right) \operatorname{sech}^{2} 2 x^{2} \end{aligned} \label{A.58} \]