# Answer this question and show the steps. sqrt{x}+sqrt{y}=3, find the value of frac{dy}{dx} at the point (4,1)

Answer this question and show the steps. $\sqrt{x}+\sqrt{y}=3$, find the value of $\frac{dy}{dx}$ at the point (4,1)
You can still ask an expert for help

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

crocolylec
Step 1 Consider the given function $\sqrt{x}+\sqrt{y}=3$ Step 2 Differentiate implicitly with respect to x. $\frac{d}{dx}\left(\sqrt{x}+\sqrt{y}\right)=\frac{d}{dx}\left(3\right)$
$\frac{d}{dx}\left({x}^{\frac{1}{2}}\right)+\frac{d}{dx}\left({y}^{\frac{1}{2}}\right)=0$
$\frac{1}{2}{x}^{\frac{1}{2}-1}+\left(\frac{1}{2}{y}^{\frac{1}{2}-1}\right)\frac{dy}{dx}=0$
$\frac{1}{2}{x}^{-\frac{1}{2}}+\left(\frac{1}{2}{y}^{-\frac{1}{2}}\right)\frac{dy}{dx}=0$
$\frac{1}{2\sqrt{x}}+\frac{1}{2\sqrt{y}}\frac{dy}{dx}=0$
$\frac{dy}{dx}=-\frac{\sqrt{y}}{\sqrt{x}}=-\sqrt{\frac{y}{x}}$ Step 3 Now, find the value of $\frac{dy}{dx}$ at the point (4, 1) $\frac{dy}{dx}\left(x,y\right)=\left(4,1\right)=-\sqrt{\frac{1}{4}}=-\frac{1}{2}$ The answer is $-\frac{1}{2}$ or -0.5(in decimals).