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Question Number 124956 by liberty last updated on 07/Dec/20

$$y={\sec }^{-1}\left(e^{x^4}\right).Finddy=?$$

Answered by bemath last updated on 07/Dec/20

$$\Rightarrow e^{x^4}=\sec yor\cos y=\frac{1}{e^{x^4}}$$$$\cos y=e^{-x^4}$$$$-\sin ydy=-4x^3{e}^{-x^4}dx$$$$dy=\frac{4x^3{e}^{-x^4}}{\sin y}dx=4x^3{e}^{-x^4}\sqrt{1-e^{-2x^4}}dx$$$$dy=\frac{4x^3}{e^{x^4}\sin \left({\sec }^{-1}\left(e^{x^4}\right)\right)}dx$$

Answered by Dwaipayan Shikari last updated on 07/Dec/20

$$secy=e^{x^4}$$$$\Rightarrow -log\left(cosy\right)=x^4$$$$\Rightarrow \frac{siny}{cosy}.\frac{dy}{dx}=4x^3\Rightarrow \frac{dy}{dx}=4x^{3}coty\Rightarrow 4x^{3}cot\left({sec}^{-1}\left(e^{x^4}\right)\right)$$

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