Implicit Differentiation problem (Exponential Derivatives) Please help!

Question

Use the process of implicit differentiation to find $dy/dx$ given that:

$$x^2e^y − y^2e^x=0 $$

I am trying first to find $y$,

$$y^2e^x = x^2e^y$$

$$y^2 = (x^2e^y)/e^x$$

$$y = \sqrt{(x^2e^y)/e^x}$$

Is this correct? I have the feeling it is not.

Answer 1

The math is right, but if you are using implicit differentiation the point is not to solve for $y$. Instead you would differentiate each term with respect to $x$, assuming that $y$ is some function of $x$ whose derivative is $dy/dx$. For instance, the term $y^2e^x$ yields

$$2y\frac{dy}{dx}e^x+y^2e^x$$

Hopefully this helps as a starting point. (You will need to differentiate the equation term by term as shown, then solve for $dy/dx$.)

Answer 2

So, implicitly differentiating; $$x^{2}\frac{dy}{dx}e^{y}+2x e^{y}-\left(y^{2}e^{x}-2y\frac{dy}{dx}e^{x}\right)=0$$ Collecting like terms $$\frac{dy}{dx}\left(x^{2}e^{y}-2ye^{x} \right)=y^{2}e^{x}-2xe^{y}$$ From which it is readily seen that $$\frac{dy}{dx}=\frac{y^{2}e^{x}-2xe^{y}}{\left(x^{2}e^{y}-2ye^{x} \right)}$$