$\frac{d}{dx} [\log (4x^3 - 2x)]$ is equal to

Analysis

The question asks for the derivative of a logarithmic function with respect to $x$. Specifically, we need to apply the rules of differentiation for logarithms.

Concept: Differentiation of Logarithmic Functions

To differentiate a function of the form $y = \ln(u)$, where $u$ is a function of $x$, we use the Chain Rule. The formula is:

In this problem, the function is $\log(4x^3 - 2x)$. In standard calculus contexts for JAMB, $\log$ without a base specified is typically treated as the natural logarithm ($\ln$).

Step-by-Step Solution

Step 1: Identify $u$ Let $u = 4x^3 - 2x$.

Step 2: Differentiate $u$ with respect to $x$ Using the power rule ($\frac{d}{dx} ax^n = anx^{n-1}$):

Step 3: Apply the Chain Rule formula Substitute $u$ and $\frac{du}{dx}$ into the formula $\frac{1}{u} \cdot \frac{du}{dx}$:

Step 4: Simplify the expression

Comparing this result to the given options, it matches Option D perfectly.

Exam Tip

Whenever you see a derivative of a logarithm, remember the "Rule of Reciprocals": The derivative is simply the derivative of the inside part divided by the inside part itself. Don't let the "log" symbol intimidate you; just focus on differentiating the expression inside the bracket and placing it over the original expression.