1. The problem is to find the first derivative of the function $$y = x \sqrt{x} + \frac{3}{x^5}$$ and verify if $$y' = \frac{3}{2} \sqrt{x} - \frac{15}{x^6}$$ is correct. 2. Recall the rules for derivatives: - The power rule: $$\frac{d}{dx} x^n = n x^{n-1}$$ - The derivative of a sum is the sum of the derivatives. 3. Rewrite the function to use exponents: $$y = x \cdot x^{\frac{1}{2}} + 3 x^{-5} = x^{1 + \frac{1}{2}} + 3 x^{-5} = x^{\frac{3}{2}} + 3 x^{-5}$$ 4. Differentiate term by term: $$y' = \frac{3}{2} x^{\frac{3}{2} - 1} + 3 \cdot (-5) x^{-5 - 1} = \frac{3}{2} x^{\frac{1}{2}} - 15 x^{-6}$$ 5. Rewrite the derivative in radical and fraction form: $$y' = \frac{3}{2} \sqrt{x} - \frac{15}{x^6}$$ 6. This matches the given derivative exactly, so the statement is True.