Question

How do you differentiate `f(x)=csc^4x-21cot^2x`?

Answer 1

`f'(x) = 2 csc^2x cot x (21 - 2csc^2 x)`

Explanation:

Given: `f(x) = csc^4 x - 21 cot^2 x`

Remember that `csc^4 x = (csc x)^4 " and " cot^2 x = (cot x)^2`

Use the Power Rule: `(u^n)' = n u^(n-1) u'`

Let `u = csc x; " " u' = -csc x cot x; " " n = 4`

Let `u = cot x; " " u' = - csc^2 x; " " n = 2`

`f'(x) = 4 csc^3 x (-csc x cot x) - 2(21) cot x (- csc^2 x)`

Simplify:
`f'(x) = -4 csc^4 x cot x + 42 csc^2 x cot x`

`f'(x) = 2 csc^2 cot x (21 - 2 csc^2 x)`