Find the derivative of the function. \[ f(x)=4 x^{-7 / 8}+8 x^{-1 / 2} \] \( f^{\prime}(x)= \)

To find the derivative of the function \( f(x) = 4x^{-7/8} + 8x^{-1/2} \), we can apply the power rule, which states that if \( f(x) = ax^n \), then \( f'(x) = a \cdot n \cdot x^{n-1} \). Calculating it step-by-step: 1. For the first term \( 4x^{-7/8} \): \[ \frac{d}{dx}(4x^{-7/8}) = 4 \cdot \left(-\frac{7}{8}\right) x^{-\frac{7}{8}-1} = -\frac{28}{8} x^{-15/8} = -\frac{7}{2} x^{-15/8} \] 2. For the second term \( 8x^{-1/2} \): \[ \frac{d}{dx}(8x^{-1/2}) = 8 \cdot \left(-\frac{1}{2}\right) x^{-\frac{1}{2}-1} = -4 x^{-3/2} \] 3. Combining both results, we get: \[ f'(x) = -\frac{7}{2} x^{-15/8} - 4x^{-3/2} \] Thus, the derivative is: \[ f'(x) = -\frac{7}{2} x^{-15/8} - 4x^{-3/2} \]