Find the derivative of $f(x)=x^{4} \ln \left(x^{3}+7\right)$ a.) $f^{\prime}(x)=4 x^{3}+\frac{3 x^{2}}{x^{3}+7}$ b.) $f^{\prime}(x)=\frac{12 x^{5}}{x^{3}+7}$ c.) $f^{\prime}(x)=4 x^{3} \ln \left(x^{3}+7\right)+\frac{x^{4}}{x^{3}+7}$ d.) $f^{\prime}(x)=4 x^{3} \ln \left(x^{3}+7\right)+\frac{3 x^{6}}{x^{3}+7}$

Step 1 ：Let's find the derivative of the function \(f(x)=x^{4} \ln \left(x^{3}+7\right)\).

Step 2 ：We can use the product rule and the chain rule to find the derivative.

Step 3 ：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.

Step 4 ：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 5 ：In this case, we can consider \(x^{4}\) as the first function and \(\ln \left(x^{3}+7\right)\) as the second function.

Step 6 ：The derivative of \(x^{4}\) is \(4x^{3}\).

Step 7 ：The derivative of \(\ln \left(x^{3}+7\right)\) can be found using the chain rule.

Step 8 ：The derivative of \(\ln(u)\) is \(\frac{1}{u}\) and the derivative of \(x^{3}+7\) is \(3x^{2}\).

Step 9 ：Therefore, the derivative of \(\ln \left(x^{3}+7\right)\) is \(\frac{3x^{2}}{x^{3}+7}\).

Step 10 ：So, the derivative of \(f(x)\) is \(4x^{3} \ln \left(x^{3}+7\right) + \frac{3x^{6}}{x^{3}+7}\).

Step 11 ：Final Answer: The derivative of the function \(f(x)=x^{4} \ln \left(x^{3}+7\right)\) is \(\boxed{4 x^{3} \ln \left(x^{3}+7\right)+\frac{3 x^{6}}{x^{3}+7}}\). Therefore, the correct option is (d).