26: suppose ( g(x) ) is a differentiable function satisfying ( g(3)=2 ) and ( g^{prime}(3)=6 ). consider the…

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) )?\n(a) ( ln (3) ) (b) ( 2 ln (3) ) (c) ( \frac{1}{3} ) (d) ( ln (12) ) (e) 3

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) )?\n(a) ( ln (3) ) (b) ( 2 ln (3) ) (c) ( \frac{1}{3} ) (d) ( ln (12) ) (e) 3

Answer

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)