# Given f(x)=\csc^{4}(5x^{2}-3x+1), find {f}'(x).

## Question:

Given {eq}f(x) = \csc^{4}(5x^{2}-3x+1) {/eq} find {eq}{f}'(x). {/eq}

## Differentiation:

In differentiation CHAIN RULE is one of the most commonly and frequently used method.

In this method we need to derivate step by step.

Chain Rule : Chain rule is defined as the differentiation of all function passing through existing function.

We have,

{eq}f(x) = \csc^{4}(5x^{2}-3x+1) {/eq}

On differentiating both sides,

{eq}f'(x) = (\csc^{4}(5x^{2}-3x+1))' \\ f'(x) = 4\csc^{3}(5x^{2}-3x+1)( csc(5x^{2}-3x+1) )' \\ f'(x) = 4\csc^{3}(5x^{2}-3x+1)( -csc(5x^{2}-3x+1)cot(5x^{2}-3x+1))(5x^{2}-3x+1)' \\ f'(x) = 4\csc^{3}(5x^{2}-3x+1)( -csc(5x^{2}-3x+1)cot(5x^{2}-3x+1))(10x - 3) \\ {/eq}

so,

{eq}\therefore \color{blue}{ f'(x) = (4\csc^{3}(5x^{2}-3x+1))( -csc(5x^{2}-3x+1)cot(5x^{2}-3x+1))(10x - 3) } {/eq}