This is a classic example of implicit differentiation.
This is almost the same as taking the derivate with only one term (frac(d)(dx)), but everytime there is a y, we write frac(dy)(dx), as we will be taking the derivative of "y with respect to x." In other words, if there is a y and an x together, we must apply the product rule (y will follow the same derivation as x).
The derivative of y^3*e^x is frac(d)(dx)(y^3*e^x)=y^3*e^x+3y^2e^xfrac(dy)(dx), by the product rule.
The derivative of -x^2 is -2x.
The derivative of 5*e^y is 5*e^y(frac(dy)(dx)), as the y is not with an x term, so we cannot treat it as a constant.
And of course, the derivative of 1 is 0.
Now we can rearrange our equation so that the frac(dy)(dx) is isolated.
y^3*e^x+3y^2e^xfrac(dy)(dx)-2x+5*e^y(frac(dy)(dx))=0
3y^2e^x(frac(dy)(dx))+5*e^y(frac(dy)(dx))=2x-y^3*e^x
frac(dy)(dx)=frac(2x-y^3*e^x)(3y^2e^x+5*e^y).
Hopefully I didn't do anything silly, and that this helps. Good luck :).
