Differentiate.
\[
f(x)=\ln \left(\frac{x^{3}-5}{x}\right)
\]
\[
f^{\prime}(x)=
\]
Final Answer: \(f^{\prime}(x)=\boxed{\frac{3x^{2}}{x^{3}-5}-\frac{1}{x}}\)
Step 1 :Given the function \(f(x)=\ln \left(\frac{x^{3}-5}{x}\right)\), we are asked to find its derivative.
Step 2 :First, we can simplify the function by splitting the natural logarithm into two parts using the properties of logarithms: \(f(x) = -\ln(x) + \ln(x^{3} - 5)\).
Step 3 :Next, we differentiate each part separately using the chain rule and the quotient rule.
Step 4 :The derivative of \(-\ln(x)\) is \(-\frac{1}{x}\) and the derivative of \(\ln(x^{3} - 5)\) is \(\frac{3x^{2}}{x^{3}-5}\).
Step 5 :Adding these two derivatives together, we find the derivative of the original function: \(f^{\prime}(x)=\frac{3x^{2}}{x^{3}-5}-\frac{1}{x}\).
Step 6 :Final Answer: \(f^{\prime}(x)=\boxed{\frac{3x^{2}}{x^{3}-5}-\frac{1}{x}}\)