Evaluate the derivative of the finction \[ f(x)=\sin ^{-1}\left(9 x^{5}\right) \]

Step 1 ：Given the function \(f(x)=\sin ^{-1}(9 x^{5})\).

Step 2 ：We need to find the derivative of this function.

Step 3 ：We can use the chain rule to find the derivative of a composite 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 4 ：In this case, the outer function is the inverse sine function and the inner function is \(9x^5\).

Step 5 ：The derivative of the inverse sine function is \(\frac{1}{\sqrt{1-x^2}}\) and the derivative of \(9x^5\) is \(45x^4\).

Step 6 ：Therefore, the derivative of the function is \(\frac{45x^4}{\sqrt{1-(9x^5)^2}}\).

Step 7 ：Simplifying the denominator, we get \(\frac{45x^4}{\sqrt{1 - 81x^{10}}}\).

Step 8 ：Final Answer: The derivative of the function \(f(x)=\sin ^{-1}(9 x^{5})\) is \(\boxed{\frac{45x^4}{\sqrt{1 - 81x^{10}}}}\).