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

Ask by West French. in the United States
Mar 11,2025

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Tutor-Verified Answer

Answer

\( h''(x) = 270x^{-7} - 210x^{-8} \)

Solution

We have the function \[ h(x) = 9x^{-5} - 5x^{-6}. \] **Step 1. Find the first derivative \( h'(x) \):** Differentiate term by term using the power rule: \[ \frac{d}{dx} \left(9x^{-5}\right) = 9 \cdot (-5)x^{-5-1} = -45x^{-6}, \] \[ \frac{d}{dx} \left(-5x^{-6}\right) = -5 \cdot (-6)x^{-6-1} = 30x^{-7}. \] Thus, the first derivative is: \[ h'(x) = -45x^{-6} + 30x^{-7}. \] **Step 2. Find the second derivative \( h''(x) \):** Differentiate \( h'(x) \) term by term: \[ \frac{d}{dx} \left(-45x^{-6}\right) = -45 \cdot (-6)x^{-6-1} = 270x^{-7}, \] \[ \frac{d}{dx} \left(30x^{-7}\right) = 30 \cdot (-7)x^{-7-1} = -210x^{-8}. \] Thus, the second derivative is: \[ h''(x) = 270x^{-7} - 210x^{-8}. \]

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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: \[ h^{\prime \prime}(x) = 270 x^{-7} - 210 x^{-8} \]

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