Find \( \frac{d y}{d x} \) 5. \( y=\sqrt{x}\left(x^{2}+3\right)^{5} \)

To find \( \frac{d y}{d x} \) for the function \( y=\sqrt{x}\left(x^{2}+3\right)^{5} \), we can apply the product rule and the chain rule. Let \( u = \sqrt{x} \) and \( v = (x^{2}+3)^{5} \). Then, we have: 1. The derivative of \( u \): \[ \frac{du}{dx} = \frac{1}{2\sqrt{x}} \] 2. The derivative of \( v \) using the chain rule: \[ \frac{dv}{dx} = 5(x^2 + 3)^4 \cdot 2x = 10x(x^2 + 3)^4 \] Now, applying the product rule \( \frac{d(uv)}{dx} = u \frac{dv}{dx} + v \frac{du}{dx} \): \[ \frac{d y}{d x} = \sqrt{x} \cdot 10x(x^2 + 3)^4 + (x^2 + 3)^5 \cdot \frac{1}{2\sqrt{x}} \] Thus, the derivative \( \frac{d y}{d x} \) simplifies to: \[ \frac{d y}{d x} = 10x\sqrt{x}(x^2 + 3)^4 + \frac{(x^2 + 3)^5}{2\sqrt{x}} \] And that's your final answer for the derivative!