## Derivative Patterns

Read this section to learn about patterns of derivatives. Work through practice problems 1-8.

Practice 1: The pattern is $\mathbf{D}\left(\mathrm{f}^{\mathrm{n}}(\mathrm{x})\right)=\mathrm{n} \mathrm{f}^{\mathrm{n}-1}(\mathrm{x}) \cdot \mathbf{D}(\mathbf{f}(\mathbf{x})) . \mathbf{D}\left(\mathrm{f}^{5}\right)=5 \mathrm{f}^{4} \mathbf{D}(\mathbf{f})$ and $\mathbf{D}\left(\mathrm{f}^{13}\right)=13 \mathrm{f}^{12} \mathbf{D}(\mathbf{f})$

Practice 2: $\quad \frac{\mathrm{d}}{\mathrm{dx}}\left(2 \mathrm{x}^{5}-\pi\right)^{2}=2\left(2 \mathrm{x}^{5}-\pi\right)^{1} \mathrm{D}\left(2 \mathrm{x}^{5}-\pi\right)=2\left(2 \mathrm{x}^{5}-\pi\right)^{1}\left(10 \mathrm{x}^{4}\right)=40 \mathrm{x}^{9}-20 \pi \mathrm{x}^{4}$

\begin{align*} \begin{aligned} &D\left(\left(x+7 x^{2}\right)^{1 / 2}\right)=\frac{1}{2}\left(x+7 x^{2}\right)^{-1 / 2} D\left(x+7 x^{2}\right)=\frac{1+14 x}{2 \sqrt{x+7 x^{2}}} \\ &D\left((\cos (x))^{4}\right)=4(\cos (x))^{3} D(\cos (x))=4(\cos (x))^{3}(-\sin (x))=-4 \cos ^{3}(x) \sin (x) \end{aligned} \end{align*}

Practice 3:

\begin{align*} \begin{aligned} &D\left(\frac{\cos (x)}{\sin (x)}\right)=\frac{\sin (x) D(\cos (x))-\cos (x) D(\sin (x))}{(\sin (x))^{2}} \\ &=\frac{\sin (x)(-\sin (x))-\cos (x)(\cos (x))}{\sin ^{2}(x)}=\frac{-\left(\sin ^{2}(x)+\cos ^{2}(x)\right)}{\sin ^{2}(x)}=\frac{-1}{\sin ^{2}(x)}=-\csc ^{2}(x) \end{aligned} \end{align*}

Practice 4:

\begin{align*} \begin{aligned} \mathbf{D}(\csc (x))=D\left(\frac{1}{\sin (x)}\right) &=\frac{\sin (x) D(1)-1 \mathbf{D}(\sin (x))}{\sin ^{2}(x)} \\ &=\frac{\sin (x)(0)-\cos (\mathbf{x})}{\sin ^{2}(x)}=-\frac{\cos (x)}{\sin (x)} \frac{1}{\sin (x)}=-\cot (x) \csc (x) \end{aligned} \end{align*}

Practice 5: $\quad \mathbf{D}\left(x^{5} \cdot \tan (x)\right)=x^{5} D(\tan (x))+\tan (x) D\left(x^{5}\right)=x^{5} \sec ^{2}(x)+\tan (x)\left(5 x^{4}\right)$

\begin{align*} \begin{aligned} &\frac{\mathbf{d}}{\mathrm{dt}}\left(\frac{\sec (\mathrm{t})}{\mathrm{t}}\right)=\frac{\mathrm{t} \mathbf{D}(\sec (\mathrm{t}))-\sec (\mathrm{t}) \mathbf{D}(\mathrm{t})}{\mathrm{t}^{2}}=\frac{\mathrm{t} \cdot \sec (\mathrm{t}) \cdot \tan (\mathrm{t})-\sec (\mathrm{t})}{\mathrm{t}^{2}} \\ &\begin{array}{l} \mathrm{D}\left((\cot (\mathrm{x})-\mathrm{x})^{1 / 2}\right) & =\frac{1}{2}(\cot (\mathrm{x})-\mathrm{x})^{-1 / 2} \mathrm{D}(\cot (\mathrm{x})-\mathrm{x}) \\ & =\frac{1}{2}(\cot (\mathrm{x})-\mathrm{x})^{-1 / 2}\left(-\csc ^{2}(\mathrm{x})-1\right)=\frac{-\csc ^{2}(\mathrm{x})-1}{2 \sqrt{\cot (\mathrm{x})-\mathrm{x}}} \end{array} \end{aligned} \end{align*}

Practice 6:

$h$ $\frac{2^{\mathrm{h}}-1}{\mathrm{~h}}$ $\frac{3^{\mathrm{h}}-1}{\mathrm{~h}}$ $\frac{e^{\mathrm{h}}-1}{\mathrm{~h}}$
0.1 0.717734625  1.16123174  1.051709181
-0.I 0.669670084  1.040415402  0.9516258196
0.01 0.69555  1.104669194  1.005016708
-0.01 0.690750451  1.092599583  0.9950166251
0.001 0.6933874  1.099215984  1.000500167
-0.001 0.69290695  1.098009035  0.9995001666

$\begin{gathered} \downarrow \\ 0 \end{gathered}$

$\begin{gathered} \downarrow \\ \approx 0.693 \end{gathered}$

$\begin{gathered} \downarrow \\ \approx 1.099 \end{gathered}$

$\begin{gathered} \downarrow \\ 1 \end{gathered}$

Practice 7: $\quad D\left(x^{3} e^{x}\right)=x^{3} D\left(e^{x}\right)+e^{x} D\left(x^{3}\right)=x^{3}\left(e^{x}\right)+e^{x}\left(3 x^{2}\right)=x^{2} \cdot e^{x} \cdot(x+3)$

\begin{align*}\begin{aligned}&D\left(\left(e^{x}\right)^{3}\right)=3\left(e^{x}\right)^{2} D\left(e^{x}\right)=3\left(e^{x}\right)^{2}\left(e^{x}\right)=3 e^{2 x} \cdot e^{x}=3 \mathrm{e}^{3 x} \end{aligned}\end{align*} or

$D\left(\left(e^{x}\right)^{3}\right)=D\left(e^{3 x}\right)=e^{3 x} D(3 x)=3 \mathbf{e}^{3 x}$

Practice 8:

$\begin{array}{l|l|l}\mathrm{f}(\mathrm{x})=3 \mathrm{x}^{7} & \mathrm{f}(\mathrm{x})=\sin (\mathrm{x}) & \mathrm{f}(\mathrm{x})=\mathrm{x} \cdot \cos (\mathrm{x}) \\\mathrm{f}^{\prime}(\mathrm{x})=21 \mathrm{x}^{6} & \mathrm{f}^{\prime}(\mathrm{x})=\cos (\mathrm{x}) & \mathrm{f}^{\prime}(\mathrm{x})=-\mathrm{x} \cdot \sin (\mathrm{x})+\cos (\mathrm{x}) \\\mathrm{f}^{\prime \prime}(\mathrm{x})=126 \mathrm{x}^{5} & \mathrm{f}^{\prime \prime}(\mathrm{x})=-\sin (\mathrm{x}) & \mathrm{f}^{\prime \prime}(\mathrm{x})=-\mathrm{x} \cdot \cos (\mathrm{x})-2 \sin (\mathrm{x}) \\\mathrm{f}^{\prime \prime \prime}(\mathrm{x})=630 \mathrm{x}^{4} & \mathrm{f}^{\prime \prime \prime}(\mathrm{x})=-\cos (\mathrm{x}) & \mathrm{f}^{\prime \prime \prime}(\mathrm{x})=\mathrm{x} \cdot \sin (\mathrm{x})-3 \cos (\mathrm{x})\end{array}$