Jason Watson

2021-12-03

Find the derivatives of the following functions with respect to x
$x=\mathrm{tan}\left({e}^{-y}\right)$

Parminquale

Step 1
Given
$x=\mathrm{tan}\left({e}^{-y}\right)$
Step 2
Differentiating wrt x
${e}^{-y}={\mathrm{tan}}^{-1}x$
Differentiating w.r.t x
${e}^{-y}\left(-\frac{dy}{dx}\right)=\frac{1}{1+{x}^{2}}$
$-{e}^{-y}\frac{dy}{dx}=\frac{1}{1+{x}^{2}}$
$\frac{dy}{dx}=\frac{-1}{\left(1+{x}^{2}\right){e}^{-y}}$
$\frac{dy}{dx}=\frac{-{e}^{y}}{1+{x}^{2}}$

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