Find the indicated derivalive for the function \( h^{\prime \prime}(x) \) for \( h(x)=9 x^{-5}-5 x^{-6} \) \( h^{\prime \prime}(x)=\square \)

To find the second derivative \( h^{\prime \prime}(x) \) for the function \( h(x)=9 x^{-5}-5 x^{-6} \), we first need to find the first derivative \( h^{\prime}(x) \). 1. First Derivative: \[ h^{\prime}(x) = \frac{d}{dx}(9 x^{-5}) - \frac{d}{dx}(5 x^{-6}) = -45 x^{-6} + 30 x^{-7} \] Now let's simplify it: \[ h^{\prime}(x) = -45 x^{-6} + 30 x^{-7} = -45x^{-6} + 30x^{-7} = -15x^{-6}(3 - 2x^{-1}) \] 2. Second Derivative: Now, take the derivative of \( h^{\prime}(x) \) to find \( h^{\prime \prime}(x) \): \[ h^{\prime \prime}(x) = \frac{d}{dx}(-45 x^{-6}) + \frac{d}{dx}(30 x^{-7}) = 270 x^{-7} - 210 x^{-8} \] Therefore, we have: \[ h^{\prime \prime}(x) = 270 x^{-7} - 210 x^{-8} \] Thus, the final answer is: \[ h^{\prime \prime}(x) = 270 x^{-7} - 210 x^{-8} \]