Find the derivative of the function. \[ f(x)=4 x^{-7 / 8}+8 x^{-1 / 2} \] \( f^{\prime}(x)=-\frac{7}{2} x^{-\left(\frac{15}{8}\right)}-4 x^{-\left(\frac{2}{3}\right)} \times \) Need Help? Read It Watch It

To find the derivative of the function \( f(x)=4 x^{-7/8}+8 x^{-1/2} \), we apply the power rule. For each term, multiply by the exponent and decrease the exponent by one. The first term, \( 4x^{-7/8} \), becomes \( 4 \cdot -\frac{7}{8} x^{-7/8-1} = -\frac{7}{2} x^{-15/8} \). The second term, \( 8x^{-1/2} \), becomes \( 8 \cdot -\frac{1}{2} x^{-1/2-1} = -4 x^{-3/2} \). So, the derivative is: \[ f'(x) = -\frac{7}{2} x^{-15/8} - 4 x^{-3/2}. \] The completion of the derivative comes with practice and mastering the rules of derivatives! Just remember to carefully apply the power rule to avoid any common mistakes, such as miscalculating the exponent. A good tip is to double-check your work step-by-step. Happy calculating!