X^(1/2)+y^(1/2)=16 can some one show me step by step because the teacher wants it written out so im trying to get it on the homework.
The equation $\displaystyle \sqrt{x}+\sqrt{y}=16$ states that the functions $\displaystyle f(x)=\sqrt{x}+\sqrt{y}$ and $\displaystyle g(x)=16$ are equal everywhere. Because they are equal, they have equal derivatives:
$\displaystyle \frac{d}{dx}(\sqrt{x})+\frac{d}{dx}(\sqrt{y})=\fra c{d}{dx}(16).$
With the Chain Rule, we can now find $\displaystyle y'$ in terms of $\displaystyle x$ and $\displaystyle y$.
$\displaystyle \sqrt x + \sqrt y = 16 \Leftrightarrow \sqrt y = 16 - \sqrt x \Rightarrow$
$\displaystyle \Rightarrow \frac{d}
{{dx}}\sqrt y = \frac{d}
{{dx}}\left( {16 - \sqrt x } \right) \Leftrightarrow \frac{{dy}}
{{dx}} \cdot \frac{1}
{{2\sqrt y }} = - \frac{1}
{{2\sqrt x }} \Leftrightarrow$
$\displaystyle \Leftrightarrow \frac{{dy}}
{{dx}} = - \frac{{\sqrt y }}
{{\sqrt x }} = - \frac{{16 - \sqrt x }}
{{\sqrt x }} = 1 - \frac{{16}}
{{\sqrt x }}.$