Find \( \frac{d y}{d x} \) for \( y=\sqrt{u} \) and \( u=x^{2}+7 \)

To find \( \frac{dy}{dx} \) for the given functions \( y = \sqrt{u} \) and \( u = x^2 + 7 \), we will use the chain rule. First, we need to find \( \frac{dy}{du} \) and \( \frac{du}{dx} \). 1. **Calculate \( \frac{dy}{du} \)**: \[ y = \sqrt{u} = u^{1/2} \] Using the power rule: \[ \frac{dy}{du} = \frac{1}{2} u^{-1/2} = \frac{1}{2\sqrt{u}} \] 2. **Calculate \( \frac{du}{dx} \)**: \[ u = x^2 + 7 \] Differentiating with respect to \( x \): \[ \frac{du}{dx} = 2x \] 3. **Apply the chain rule**: The chain rule states that: \[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \] Substituting the derivatives we found: \[ \frac{dy}{dx} = \frac{1}{2\sqrt{u}} \cdot 2x \] 4. **Substitute \( u \) back in**: Since \( u = x^2 + 7 \), we substitute this back into our expression: \[ \frac{dy}{dx} = \frac{1}{2\sqrt{x^2 + 7}} \cdot 2x = \frac{x}{\sqrt{x^2 + 7}} \] Thus, the final result is: \[ \frac{dy}{dx} = \frac{x}{\sqrt{x^2 + 7}} \]