`y = cos(sqrt(sin(tan(pix))))` Find the derivative of the function.

Expert Answers
Neethu eNotes educator| Certified Educator

Given the function `y=cos(sqrt(sin(tan(pi x))))` . We have to find the derivative.

Let us begin,

`(dy)/(dx)=-sin(sqrt(sin(tan(pi x)))).d/(dx)[sqrt(sin(tan(pi x)))] ------>(1)`


`d/(dx)[sqrt(sin(tan(pi x)))]=1/(2sqrt(sin(tan(pi x)))).d/(dx)[sin(tan(pi x))] ----->(2)`


`d/(dx)[sin(tan(pi x))]=cos(tan(pi x))d/(dx)[tan(pi x)]-------->(3)`

`=cos(tan(pi x)). pi sec^2(pi x)`

Now substituting this in (2) we get,

`d/(dx)[sqrt(sin(tan(pi x)))]=1/(2sqrt(sin(tan(pi x)))).cos(tan(pi x)).pi sec^2(pi x)`

Substituting this in (1) we get,

`(dy)/(dx)[cos(sqrt(sin(tan(pi x))))]=-sin(sqrt(sin(tan(pi x)))).pi/(2sqrt(sin(tan(pi x)))).cos(tan(pi x))sec^2(pi x)`