Question

I am needing help with a calculus I derivative question. pic included.

Answer 1

`\begin{equation*} -2csc^2(\sin x)\cos^2(\cos x)cot(\sin x)\cos x+csc^2(\sin x)\sin(2\cos x)\sin x \end{equation*}`

Step-by-step explanation:

Given:

`f(x)=(csc^2(sinx))/(sec^2(cosx))`

To find:

The derivative of f(x)

If we simplify the given function, we'll have;

`\begin{gathered} f(x)=(csc^2(sinx))/((1)/(\cos^2)(\cos x)) \\ f(x)=csc^2(\sin x)\cos^2(\cos x) \end{gathered}`

We'll go ahead and apply the product rule to determine the derivative of f(x);

`Let\text{ }u=csc^2(\sin x),\text{ }v=\cos^2(\cos x)`
`\begin{gathered} f^(\prime)(x)=u^(\prime)v+v^(\prime)u \\ \\ =[-2csc^2(\sin x)cot(\sin x)\cos x][\cos^2(\cos x)]+[\sin(2\cos x)\sin x][csc^2(\sin x)] \\ \\ =-2csc^2(\sin x)\cos^2(\cos x)cot(\sin x)\cos x+csc^2(\sin x)\sin(2\cos x)\sin x \end{gathered}`