Question

write the function below in the form ( y = f(u) ) and ( u = g(x) ), then find ( \frac{dy}{dx} ) as a function of ( x ).

( y=cot (cos x) )

what are the functions ( f(u) ) and ( g(x) )?

( f(u)=)

( g(x)=)

Explanation:

Step1: Identify inner and outer functions

Let \(u = g(x)=\cos x\). Then \(y = f(u)=\cot u\).

Step2: Find derivatives of \(f(u)\) and \(g(x)\)

The derivative of \(y = f(u)=\cot u\) with respect to \(u\) is \(f^{\prime}(u)=-\csc^{2}u\).
The derivative of \(u = g(x)=\cos x\) with respect to \(x\) is \(g^{\prime}(x)=-\sin x\).

Step3: Apply the chain rule \(\frac{dy}{dx}=f^{\prime}(u)\cdot g^{\prime}(x)\)

Substitute \(u = \cos x\), \(f^{\prime}(u)=-\csc^{2}u\) and \(g^{\prime}(x)=-\sin x\) into the chain - rule formula.
\(\frac{dy}{dx}=(-\csc^{2}(\cos x))\cdot(-\sin x)=\sin x\csc^{2}(\cos x)\)

Answer:

\(f(u)=\cot u\), \(g(x)=\cos x\), \(\frac{dy}{dx}=\sin x\csc^{2}(\cos x)\)