Diffrentiate the following w.r.t.x [ ( e ^ 2 ) ] - Vidyadip

Question

Diffrentiate the following w.r.t.x

$\log \left[\sec \left( e ^{\times 2}\right)\right]$

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Answer

Let $y=\log \left[\sec \left(e^{x 2}\right)\right]$

Differentiating w.r.t. x, we get

$\begin{aligned} \frac{d y}{d x} & =\frac{d}{d x}\left[\log \left(\sec e^{x^2}\right)\right] \\ & =\frac{1}{\sec \left(e^{x^2}\right)} \cdot \frac{d}{d x}\left[\sec \left(e^{x^2}\right)\right] \\ & =\frac{1}{\sec \left(e^{x^2}\right)} \cdot \sec \left(e^{x^2}\right) \tan \left(e^{x^2}\right) \cdot \frac{d}{d x}\left(e^{x^2}\right) \\ & =\tan \left(e^{x^2}\right) \cdot e^{x^2} \cdot \frac{d}{d x}\left(x^2\right) \\ & =\tan \left(e^{x^2}\right) \cdot e^{x^2} \cdot 2 x \\ & =2 x \cdot e^{x^2} \tan \left(e^{x^2}\right) .\end{aligned}$

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