Question

let ( f ) be a differentiable function. if ( h(x)=(1 + f(3x))^{2} ), which of the following gives a correct process for finding ( h(x) )?

a ( h(x)=2(1 + f(3x)) )
b ( h(x)=2(1 + f(3x))cdot f(3x) )
c ( h(x)=2(1 + f(3x))cdot f(x) )
d ( h(x)=2(1 + f(3x))cdot f(3x)cdot 3 )

Explanation:

Step1: Apply the chain rule

Let \(u = 1 + f(3x)\), then \(h(x)=u^{2}\). The derivative of \(h(x)\) with respect to \(u\) is \(h^\prime(u) = 2u\).

Step2: Differentiate \(u\) with respect to \(x\)

The derivative of \(u = 1 + f(3x)\) with respect to \(x\): The derivative of \(1\) is \(0\). For \(y = f(3x)\), let \(t = 3x\), then \(y = f(t)\). By the chain rule, \(\frac{dy}{dx}=\frac{dy}{dt}\cdot\frac{dt}{dx}\). \(\frac{dt}{dx}=3\) and \(\frac{dy}{dt}=f^\prime(t)\), so \(\frac{dy}{dx}=3f^\prime(3x)\).

Step3: Combine using the chain rule

By the chain rule \(\frac{dh}{dx}=\frac{dh}{du}\cdot\frac{du}{dx}\). Substituting \(u = 1 + f(3x)\) and \(\frac{du}{dx}=3f^\prime(3x)\) into \(\frac{dh}{dx}\), we get \(h^\prime(x)=2(1 + f(3x))\cdot3f^\prime(3x)\)

Answer:

D. \(h^\prime(x)=2(1 + f(3x))\cdot f^\prime(3x)\cdot3\)