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

Answers (1)

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

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