Question

26: suppose ( g(x) ) is a differentiable function satisfying ( g(3)=2 ) and ( g^{prime}(3)=6 ). consider the function ( f(x)=ln (g(x)) ). which of the following is equal to ( f^{prime}(3) )?
(a) ( ln (3) ) (b) ( 2 ln (3) ) (c) ( \frac{1}{3} ) (d) ( ln (12) ) (e) 3

Explanation:

Step1: Apply the chain rule

The chain rule states that if \(y = f(u)\) and \(u = g(x)\), then \(y^\prime=\frac{dy}{du}\cdot\frac{du}{dx}\). For \(f(x)=\ln(g(x))\), let \(u = g(x)\), then \(f^\prime(x)=\frac{1}{g(x)}\cdot g^\prime(x)\) (since \(\frac{d}{du}(\ln u)=\frac{1}{u}\)).

Step2: Evaluate at \(x = 3\)

Substitute \(x = 3\) into \(f^\prime(x)\). We know that \(g(3)=2\) and \(g^\prime(3)=6\). So \(f^\prime(3)=\frac{g^\prime(3)}{g(3)}\).

Step3: Calculate the value

Substitute the values of \(g(3)\) and \(g^\prime(3)\) into the formula. \(f^\prime(3)=\frac{6}{2}=3\).

Answer:

\(e.3\)