Question

given ( y = f(u) ) and ( u = g(x) ), find ( \frac{dy}{dx} = f(g(x))g(x) ) for the following functions.

( y=cos u, u = 6x + 5 )

( \frac{dy}{dx}=f(g(x))g(x)= )

Explanation:

Step1: Find the derivative of \(y = f(u)\)

Given \(y=\cos u\), the derivative \(f^{\prime}(u)=-\sin u\).

Step2: Substitute \(u = g(x)\) into \(f^{\prime}(u)\)

Since \(u = 6x + 5\), then \(f^{\prime}(g(x))=-\sin(6x + 5)\).

Step3: Find the derivative of \(u = g(x)\)

Given \(u = 6x+5\), the derivative \(g^{\prime}(x)=6\).

Step4: Calculate \(\frac{dy}{dx}=f^{\prime}(g(x))g^{\prime}(x)\)

Using the chain - rule formula \(\frac{dy}{dx}=f^{\prime}(g(x))g^{\prime}(x)\), we substitute \(f^{\prime}(g(x))=-\sin(6x + 5)\) and \(g^{\prime}(x)=6\) into it. So \(\frac{dy}{dx}=-\sin(6x + 5)\times6=-6\sin(6x + 5)\)

Answer:

\(-6\sin(6x + 5)\)