1. **State the problem:** Differentiate the function $$f(x) = 4x^{\frac{1}{2}} + \frac{3}{2}x^{-\frac{5}{8}}$$ with respect to $$x$$. 2. **Recall the power rule for differentiation:** For any term $$ax^b$$, the derivative is $$f'(x) = abx^{b-1}$$. 3. **Apply the power rule to each term:** - For $$4x^{\frac{1}{2}}$$, the derivative is $$4 \times \frac{1}{2} x^{\frac{1}{2} - 1} = 2x^{-\frac{1}{2}}$$. - For $$\frac{3}{2}x^{-\frac{5}{8}}$$, the derivative is $$\frac{3}{2} \times \left(-\frac{5}{8}\right) x^{-\frac{5}{8} - 1} = -\frac{15}{16} x^{-\frac{13}{8}}$$. 4. **Combine the derivatives:** $$f'(x) = 2x^{-\frac{1}{2}} - \frac{15}{16} x^{-\frac{13}{8}}$$ 5. **Final answer:** $$\boxed{f'(x) = 2x^{-\frac{1}{2}} - \frac{15}{16} x^{-\frac{13}{8}}}$$