Answer:
the question is incomplete, the complete question is
"Finding Derivatives Implicity In Exercise,Find dy/dx implicity . [tex]x^{2}e^{-x}+2y^{2}-xy[/tex]"
Answer : [tex]\frac{dy}{dx}=\frac{y-(2-x)xe^{-x}}{(4y-x)}[/tex]
Step-by-step explanation:
From the expression [tex]x^{2}e^{-x}+2y^{2}-xy[/tex]" y is define as an implicit function of x, hence we differentiate each term of the equation with respect to x.
we arrive at
[tex]\frac{d}{dx}(x^{2}e^{-x )+\frac{d}{dx} (2y^{2})-\frac{d}{dx}xy=0\\[/tex]
for the expression [tex]\frac{d}{dx}(x^{2}e^{-x})[/tex] we differentiate using the product rule, also since y^2 is a function of y which itself is a function of x, we have
[tex](2xe^{-x}-x^{2}e^{-x})+4y\frac{dy}{dx}-x\frac{dy}{dx} -y=0\\\\(2-x)xe^{-x}+(4y-x)\frac{dy}{dx}-y=0 \\[/tex].
if we make dy/dx subject of formula we arrive at
[tex](4y-x)\frac{dy}{dx}=y-(2-x)xe^{-x}\\\frac{dy}{dx}=\frac{y-(2-x)xe^{-x}}{(4y-x)}[/tex]
