The correct answer is Question 2 Select the best answer in computing the derivative of $k(x)=6 x^{4} \ln \left(4 x^{2}+7\right)$. a. $24 x^{3} \ln \left(4 x^{2}+7\right)+6 x^{4} \cdot \frac{8 x}{4 x^{2}+7}$ b. $24 x^{3} \cdot \frac{1}{8 x}$ c. $24 x^{3} \ln (8 x) \cdot \frac{8 x}{4 x^{2}+7}$ d. $24 x^{3} \ln \left(4 x^{2}+7\right) \cdot \frac{8 x}{4 x^{2}+7}$ e. None of these are correct. The best answer is

Step 1 ：The question is asking for the derivative of the function \(k(x)=6 x^{4} \ln \left(4 x^{2}+7\right)\). To solve this, we need to use the product rule and the chain rule of differentiation. The product rule states that the derivative of a product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function. The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function.

Step 2 ：The derivative of \(x^{4}\) is \(4x^{3}\) and the derivative of \(\ln \left(4 x^{2}+7\right)\) is \(\frac{1}{4 x^{2}+7} \cdot 8x\) by the chain rule.

Step 3 ：So, the derivative of \(k(x)\) is \(6 \left[4x^{3} \ln \left(4 x^{2}+7\right) + x^{4} \cdot \frac{8x}{4 x^{2}+7}\right]\).

Step 4 ：Let's simplify this expression and compare it with the given options. The derivative of the function \(k(x)\) is \(48x^{5}/(4x^{2} + 7) + 24x^{3} \ln(4x^{2} + 7)\). This expression can be rewritten as \(24x^{3} \ln(4x^{2} + 7) + 6x^{4} \cdot \frac{8x}{4x^{2} + 7}\), which matches option a.

Step 5 ：Final Answer: The best answer is \(\boxed{\text{(a)}\ 24x^{3} \ln(4x^{2} + 7) + 6x^{4} \cdot \frac{8x}{4x^{2} + 7}}\).