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Differentiate log sin (x^2) with respect to x.

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Solution: We have , $y=log (sin x^2)$ putting $v=x^2$ and $u=sin x^2=sin v,$ we get

$y=log (u),$ $u=sin v$ and $v=x^2$

$herefore$ $fracmathrmd ymathrmd x=frac1u$ , $fracmathrmd umathrmd v=cosv$ and $fracmathrmd vmathrmd x=2x.$

Now , $fracmathrmd ymathrmd x=fracmathrmd ymathrmd u imes fracmathrmd umathrmd v imes fracmathrmd vmathrmd x$

$Rightarrow$ $fracmathrmd ymathrmd x=frac1u imes cos v imes 2x=frac1sinvcosv imes2x$ $[ecause u=sinv]$

$Rightarrow$ $fracmathrmd ymathrmd x=(cotv)2x=2xcot(x^2)$ $[ecause v=x^2]$

Hence , $fracmathrmd (log(sinx^2))mathrmd x=2xcot x^2$

Posted by

Deependra Verma

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