Let $x$ represent units of labor and $y$ the capital invested in a manufacturing process. When $213,415$ units are produced, the relation between labor and capital can be modeled by:
$140x^{.75}y^{.25}=213,415$
Find the rate of change of $x$ with respect to $y$ when $x=3000$ and $y=200$.
$solution:$
We will implicitly differentiate, solve for $\frac{dy}{dx}$, and then plug in $x=3000$ and $y=200$
$\frac{d}{dx}140x^{.75}y^{.25}=\frac{d}{dx}213,415=0$
$140\frac{d}{dx}(x^{.75}y^{.25})=0$
No we we have to use the product rule to evaluate $\frac{d}{dx}(x^{.75}y^{.25})$ as so:
$\frac{d}{dx}(x^{.75}y^{.25})$
$= (\frac{d}{dx}x^{.75})y^{.25}+x^{.75}(\frac{d}{dx}y^{.25})$
$=.75x^{-.25}y^{.25}+x^{.75}(.25y^{-.75} \frac{dy}{dx})$
And thus:
$140(.75x^{-.25}y^{.25}+x^{.75}(.25y^{-.75} \frac{dy}{dx}))=0$
$140(.75x^{-.25}y^{.25}+.25x^{.75}y^{-.75}\frac{dy}{dx})=0$
$105x^{-.25}y^{.25}+35x^{.75}y^{-.75}\frac{dy}{dx}=0$
$\rightarrow$
$35x^{.75}y^{-.75}\frac{dy}{dx}=-105x^{-.25}y^{.25}$
$\frac{dy}{dx} = \frac{-105x^{-.25}y^{.25}}{35x^{.75}y^{-.75}}$
Noting that $x^{-.25} = \frac{1}{x^{.25}}$ and $\frac{1}{y^{-.75}}=y^{.75}$ we see that:
$\frac{dy}{dx} = \frac{-3y}{x}$
Plugging in $y=200$, $x=3000$, we get:
$\frac{dy}{dx} = -.2$
HINT.-$f(x,y)=Ax^{75}y^{25}-B=0$ and $$\dfrac{dx}{dy}=-\dfrac{\dfrac{\partial f}{\partial y}}{\dfrac{\partial f}{\partial y}}=-\dfrac{x^{75}y^{24}}{3x^{74}y^{25}}=\dfrac{-x}{3y}$$