calculus question

Question

Write the composite function in the form f(g(x)). [Identify the inner function u =g(x) and the outer function y =f (u).] Then find the derivative dy/dx.

y=sin(cot x)

Answer 1

Let g(x) = cotx and f(x) = sinx

Then f(g(x)) = f(cotx) = sin(cotx).

If y = f(u), where u is a function of x, then, by the Chain Rule, dy/dx = (dy/du)(du/dx)

So, if u = cotx, and y = f(x) = sinx, then y = f(u) = sinu

f'(cotx) = (dy/du)(du/dx) = (cosu)(-csc²x) = cos(cotx)(-csc²x)