1. **State the problem:** Given the function $$f(x) = \frac{5}{4\sqrt{x}} - \frac{\sqrt{x^3}}{4}$$, find the derivative $$f'(x)$$ and then evaluate $$f'(1)$$. Express the answer as a single simplified fraction. 2. **Rewrite the function with exponents:** Recall that $$\sqrt{x} = x^{1/2}$$ and $$\sqrt{x^3} = (x^3)^{1/2} = x^{3/2}$$. So, $$f(x) = \frac{5}{4x^{1/2}} - \frac{x^{3/2}}{4} = \frac{5}{4}x^{-1/2} - \frac{1}{4}x^{3/2}$$ 3. **Differentiate term-by-term using the power rule:** - For $$\frac{5}{4}x^{-1/2}$$, derivative is $$\frac{5}{4} \cdot (-\frac{1}{2}) x^{-3/2} = -\frac{5}{8} x^{-3/2}$$ - For $$-\frac{1}{4}x^{3/2}$$, derivative is $$-\frac{1}{4} \cdot \frac{3}{2} x^{1/2} = -\frac{3}{8} x^{1/2}$$ So, $$f'(x) = -\frac{5}{8} x^{-3/2} - \frac{3}{8} x^{1/2}$$ 4. **Evaluate at $$x=1$$:** - $$x^{-3/2} = 1^{-3/2} = 1$$ - $$x^{1/2} = 1^{1/2} = 1$$ Therefore, $$f'(1) = -\frac{5}{8} - \frac{3}{8} = -\frac{8}{8} = -1$$ 5. **Final answer:** $$f'(1) = -1$$