Consider the following. \[ \sqrt{x}+\sqrt{y}=4 \] (a) Find \( y^{\prime} \) by implict differentiation. (b) Solve the equation explicitly for \( y \) and differentiate to get \( y^{\prime} \) In terms of \( x \). \( y^{\prime}=\square \) (c) Check that your solutions to parts (a) and (b) are consistent by substituting the expression for \( y \) into your solution for part (a), \( y^{\prime}=\square \) Need Heln?

To find \( y' \) through implicit differentiation, we start with the equation: \[ \sqrt{x}+\sqrt{y}=4 \] Differentiating both sides with respect to \( x \): \[ \frac{1}{2\sqrt{x}} + \frac{1}{2\sqrt{y}} \cdot \frac{dy}{dx} = 0 \] Now, isolate \( \frac{dy}{dx} \): \[ \frac{1}{2\sqrt{y}} \cdot \frac{dy}{dx} = -\frac{1}{2\sqrt{x}} \] Multiplying both sides by \( 2\sqrt{y} \) gives: \[ \frac{dy}{dx} = -\frac{\sqrt{y}}{\sqrt{x}} \] For part (b), we solve for \( y \) from the original equation: \[ \sqrt{y} = 4 - \sqrt{x} \implies y = (4 - \sqrt{x})^2 \] Now we differentiate \( y \): \[ y' = \frac{d}{dx}((4 - \sqrt{x})^2) = 2(4 - \sqrt{x})(-\frac{1}{2\sqrt{x}}) = -\frac{(4 - \sqrt{x})}{\sqrt{x}} \] So, in terms of \( x \): \[ y' = -\frac{4 - \sqrt{x}}{\sqrt{x}} \] For part (c), we substitute \( y = (4 - \sqrt{x})^2 \) into the expression from part (a): \[ y' = -\frac{\sqrt{(4 - \sqrt{x})^2}}{\sqrt{x}} = -\frac{(4 - \sqrt{x})}{\sqrt{x}} \] Thus, the expressions from parts (a) and (b) are consistent. Therefore, the value of \( y' \) from both methods is \( y^{\prime} = -\frac{4 - \sqrt{x}}{\sqrt{x}} \).