We need to differentiate $\log(4x^3 - 2x)$ with respect to $x$. Recall that the derivative of $\log u$ is $\frac{1}{u} \cdot \frac{du}{dx}$, by the chain rule.

Here $u = 4x^3 - 2x$. First find $\frac{du}{dx}$: $\frac{d}{dx}(4x^3) = 12x^2$ and $\frac{d}{dx}(-2x) = -2$, so $\frac{du}{dx} = 12x^2 - 2$.

Then $\frac{d}{dx} \log(4x^3 - 2x) = \frac{1}{4x^3 - 2x} \cdot (12x^2 - 2) = \frac{12x^2 - 2}{4x^3 - 2x}$.

Option D matches this exactly. Option A has denominator $4x^2$ instead of $4x^3 - 2x$, which is incorrect. Options B and C have unrelated expressions.

So the correct answer is D) $\frac{12x^2 - 2}{4x^3 - 2x}$.

Textbook references:

• Essential Mathematics for Senior Secondary Schools, p. 689
• Nelson Functional Mathematics 3 for Senior Secondary Schools, p. 211