djeljenike

2021-05-30

Find the derivatives of the given functions. $z\left(x\right)=\mathrm{ln}|\mathrm{sec}x+\mathrm{tan}x|$

Ayesha Gomez

$z\left(x\right)=\mathrm{ln}|\mathrm{sec}x+\mathrm{tan}x|$
Differentiate both sides with respect to x
${z}^{\prime }\left(x\right)=\frac{d}{dx}\left[\mathrm{ln}|\mathrm{sec}x+\mathrm{tan}x|\right]$
Apply $\frac{d}{dx}\left[\mathrm{ln}|u|\right]=\frac{1}{u}\frac{du}{dx}$
${z}^{\prime }\left(x\right)=\frac{1}{\mathrm{sec}x+\mathrm{tan}xdx}\frac{d}{dx}\left[\mathrm{sec}x+\mathrm{tan}x\right]$
Therefore,
${z}^{\prime }\left(x\right)=\frac{1}{\mathrm{sec}x+\mathrm{tan}x}\left(\mathrm{sec}x\mathrm{tan}x+{\mathrm{sec}}^{2}x\right)$
Simplify
${z}^{\prime }\left(x\right)=\frac{\mathrm{sec}x}{\mathrm{sec}x+\mathrm{tan}x}\left(\mathrm{tan}x+\mathrm{sec}x\right)$
${z}^{\prime }\left(x\right)=\mathrm{sec}x$

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