Question

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

Answer 1

ANSWER:

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

EXPLANATION:

Given:

$f(x)=\frac{csc^2(sinx)}{sec^2(cosx)}$

To find:

The derivative of f(x)

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

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

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}$