Using logarithmic differentiation, find the derivative of $y=\frac{x^{4} \sqrt[3]{x^{5}-3}}{x^{2}-25}$
a.) $\frac{d y}{d x}=\frac{x^{4} \sqrt[3]{x^{5}-3}}{x^{2}-25}\left(\frac{4}{x}+\frac{5 x^{4}}{3\left(x^{5}-3\right)}-\frac{2 x}{x^{2}-25}\right)$
b.) $\frac{d y}{d x}=\frac{4}{x}+\frac{5 x^{4}}{3\left(x^{5}-3\right)}+\frac{2 x}{x^{2}-25}$
c.) $\frac{d y}{d x}=\frac{4}{x}+\frac{5 x^{4}}{3\left(x^{5}-3\right)}-\frac{2 x}{x^{2}-25}$
d.) $\frac{d y}{d x}=\frac{x^{4} \sqrt[3]{x^{5}-3}}{x^{2}-25}\left(\frac{4}{x}+\frac{5 x^{4}}{3\left(x^{5}-3\right)}+\frac{2 x}{x^{2}-25}\right)$

Step 1 ：Take the natural logarithm of both sides of the equation: \(\ln y = \ln\left(\frac{x^{4} \sqrt[3]{x^{5}-3}}{x^{2}-25}\right)\)

Step 2 ：Use the properties of logarithms to simplify the equation: \(\ln y = 4\ln x - \ln(x^2 - 25) + \frac{1}{3}\ln(x^5 - 3)\)

Step 3 ：Differentiate both sides of the equation with respect to x: \(\frac{d\ln y}{dx} = \frac{5x^4}{3(x^5 - 3)} - \frac{2x}{x^2 - 25} + \frac{4}{x}\)

Step 4 ：Solve the equation for y': \(\frac{dy}{dx} = \frac{x^{4} \sqrt[3]{x^{5}-3}}{x^{2}-25}\left(\frac{4}{x}+\frac{5 x^{4}}{3\left(x^{5}-3\right)}-\frac{2 x}{x^{2}-25}\right)\)

Step 5 ：Final Answer: \(\boxed{\frac{x^{4} \sqrt[3]{x^{5}-3}}{x^{2}-25}\left(\frac{4}{x}+\frac{5 x^{4}}{3\left(x^{5}-3\right)}-\frac{2 x}{x^{2}-25}\right)}\)