Questions: If g'(x) = 2x^2, then the first derivative of ln(g(x)) equals A. 4x B. 4x / g(x) C. 2x^2 / g(x) D. ln(2x^2) E. None of the above.

Transcript text: If $g^{\prime}(x)=2 x^{2}$, then the first derivative of $\ln (g(x))$ equals A. $4 x$ B. $4 x / g(x)$ C. $2 x^{2} / g(x)$ D. $\ln \left(2 x^{2}\right)$ E. None of the above.

Solution

Solution Steps

To find the first derivative of $\ln(g(x))$ given that $g'(x) = 2x^2$, we can use the chain rule. The chain rule states that if $h(x) = \ln(g(x))$, then $h'(x) = \frac{g'(x)}{g(x)}$. Substituting $g'(x) = 2x^2$ into this formula will give us the desired derivative.

Step 1: Given Information

We are given that $ g'(x) = 2x^2 $ and we need to find the first derivative of $ \ln(g(x)) $.

Step 2: Use the Chain Rule

To find the derivative of $ \ln(g(x)) $, we use the chain rule. The chain rule states that if $ h(x) = \ln(g(x)) $, then: \[ h'(x) = \frac{d}{dx} \ln(g(x)) = \frac{g'(x)}{g(x)} \]

Step 3: Substitute $ g'(x) $

We know from the problem that $ g'(x) = 2x^2 $. Substituting this into the chain rule formula, we get: \[ h'(x) = \frac{2x^2}{g(x)} \]