Ramsey

2021-06-19

Find the derivatives of the given functions. $w\left(x\right)=\mathrm{sec}x\mathrm{tan}\left({x}^{2}-1\right)$

lamusesamuset

$w\left(x\right)=\mathrm{sec}x\mathrm{tan}\left({x}^{2}-1\right)$
${w}^{\prime }\left(x\right)=\mathrm{sec}x\stackrel{˙}{\left\{}\frac{d}{dx}\left(\mathrm{tan}\left({x}^{2}-1\right)+\mathrm{tan}\left({x}^{2}-1\right)\stackrel{˙}{\left\{}\frac{d}{dx}\left(\mathrm{sec}x\right)$
$=\mathrm{sec}x\stackrel{˙}{\left\{}\left({x}^{2}-1\right)\stackrel{˙}{\left\{}\frac{d}{dx}\left({x}^{2}-1\right)+\mathrm{tan}\left({x}^{2}-1\right)\stackrel{˙}{\left\{}\mathrm{sec}x\stackrel{˙}{\mathrm{tan}x}$
$==\mathrm{sec}x\stackrel{˙}{\left\{}{\mathrm{sec}}^{2}\left({x}^{2}-1\right)\stackrel{˙}{\left\{}2x+\mathrm{sec}x\stackrel{˙}{\left\{}\mathrm{tan}x\stackrel{˙}{\left\{}\mathrm{tan}\left({x}^{2}-1\right)$
$=2x\stackrel{˙}{\left\{}\mathrm{sec}x\stackrel{˙}{\left\{}{\mathrm{sec}}^{2}\left({x}^{2}-1\right)+\mathrm{sec}x\stackrel{˙}{\left\{}\mathrm{tan}x\stackrel{˙}{\left\{}\mathrm{tan}\left({x}^{2}-1\right)$

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