Question

How do you differentiate `f(x) = (cosx)/(sinx)` using the quotient rule?

Answer 1

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

Explanation:

Use the quotient rule.

Let `f(x) = (g(x))/(h(x))`, then `g(x) = cosx` and `h(x) = sinx`. Then `g'(x) = -sinx` and `h'(x) = cosx`.

`f'(x) = (g'(x) * h(x) - g(x) * h'(x))/(h(x))^2`

`f'(x) = (-sinx(sinx) - cosx(cosx))/(sinx)^2`

`f'(x)= (-sin^2x - cos^2x)/sin^2x`

`f'(x) = (-(sin^2x + cos^2x))/sin^2x`

`f'(x) = -1/sin^2x`

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

Hopefully this helps!