Answer
$(a)$ $$\frac{dy}{dx}=\frac{2x^{-1/2}}{-y^{-1/2}}$$
$(b)$ The slope of the tangent line to the curve is $m=-1$
Work Step by Step
$(a)$ Use implicit differentiation to determine $\frac{dy}{dx}$ for $\sqrt{x}-2\sqrt{y}=0$
Taking the derivative implicitly we get:
$$\frac{1}{2}x^{-1/2}-y^{-1/2}\frac{dy}{dx}=0$$
solve for $\frac{dy}{dx}$
$$\frac{dy}{dx}=\frac{2x^{-1/2}}{-y^{-1/2}}$$
$(b)$ Find the slope of the tangent line to the curve at $(4,1)$
We plug in the point $(4,1)$ into the derivative from part $(a)$. Hence:
$$\frac{dy}{dx}=\frac{2(4)^{-1/2}}{-(1)^{-1/2}}=\frac{2(1/2)}{-1}=\frac{1}{-1}=-1$$
