If y = ^ -1 [ (4^ x ) ], find d y d x - Vidyadip

Question

If $y =\cos ^{-1}\left[\sin \left(4^{ x }\right)\right]$, find $\frac{ d y}{ d x}$

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Answer

$ y=\cos ^{-1}\left[\sin \left(4^x\right)\right]$
$=\cos ^{-1}\left[\cos \left(\frac{\pi}{2}-4^x\right)\right]$
$=y=\frac{\pi}{2}-4^x $
Differentiating w.r.t. $x$, we get
$ \frac{ d y}{ d x}=\frac{ d }{ d x}\left(\frac{\pi}{2}-4^x\right)$
$=0-4^{ x } \log 4$
$=-4^{ x } \log 4 $

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