If `y=log tan.(x)/(2)`, find `(dy)/(dx)`.

1.	If `y=log tan.(x)/(2)`, find `(dy)/(dx)`.
Answer» Given: `y=log tan.(x)/(2)`. Putting `(x)/(2)=t and tan.(x)/(2)=tant=u`, we get `y=logu, u = tan t and t=(x)/(2)` `rArr(dy)/(dx)=(1)/(u),(du)/(dt)=sec^(2)t and (dt)/(dx)=(1)/(2)` `rArr(dy)/(dx)=((dy)/(du)xx(du)/(dt)xx(dt)/(dx))` `=((1)/(u)xxsec^(2)txx(1)/(2))=((1)/(2tant)xxsec^(2)t)=(1)/(sin2t)=(1)/(sinx)="cosec x".` Hence, `(dy)/(dx)="cosec x"`.

Discussion

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