y-sec-1-e-x-4-Find-dy-

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

y = \sec^{- 1} (e^{x^{4}}) . F i n d d y = ?

Answered by bemath last updated on 07/Dec/20

\Rightarrow e^{x^{4}} = \sec y o r \cos y = \frac{1}{e^{x^{4}}}

\cos y = e^{- x^{4}}

- \sin y d y = - 4 x^{3} e^{- x^{4}} d x

d y = \frac{4 x^{3} e^{- x^{4}}}{\sin y} d x = 4 x^{3} e^{- x^{4}} \sqrt{1 - e^{- 2 x^{4}}} d x

d y = \frac{4 x^{3}}{e^{x^{4}} \sin (\sec^{- 1} (e^{x^{4}}))} d x

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

s e c y = e^{x^{4}}

\Rightarrow - l o g (c o s y) = x^{4}

\Rightarrow \frac{s i n y}{c o s y} . \frac{d y}{d x} = 4 x^{3} \Rightarrow \frac{d y}{d x} = 4 x^{3} c o t y \Rightarrow 4 x^{3} c o t ({s e c}^{- 1} (e^{x^{4}}))