5. Given $y = x\sqrt{x}$, find $\frac{d^2y}{dx^2}$. Step 1: Rewrite the function $y$ in exponential form. $$ y = x \cdot x^{1/2} = x^{1 + 1/2} = x^{3/2} $$ Step 2: Find the first derivative $\frac{dy}{dx}$. Apply the power rule $\frac{d}{dx}(x^n) = nx^{n-1}$. $$ \frac{dy}{dx} = \frac{3}{2}x^{\frac{3}{2}-1} = \frac{3}{2}x^{1/2} $$ Step 3: Find the second derivative $\frac{d^2y}{dx^2}$. Differentiate $\frac{dy}{dx}$ again using the power rule. $$ \frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{3}{2}x^{1/2}\right) = \frac{3}{2} \cdot \frac{1}{2}x^{\frac{1}{2}-1} $$ $$ \frac{d^2y}{dx^2} = \frac{3}{4}x^{-1/2} $$ Step 4: Express the result with a positive exponent. $$ \frac{d^2y}{dx^2} = \frac{3}{4\sqrt{x}} $$ The final answer is $\boxed{\frac{3}{4\sqrt{x}}}$. 3 done, 2 left today. You're making progress.