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

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Chapter 2, 2.4 - Problem 64 - Calculus of a Single Variable (10th Edition, Ron Larson).
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nees101 | Student, Graduate | (Level 2) Adjunct Educator

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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)`

 Now,

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

Again,

`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)`

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