Technique: chain rule and product rule.
Application:
Recall for any variable y we have a formula that gives the derivative of y^n with respect to y as n*y^(n-1). It turns out for taking derivatives of an expression containing some variable with respect to a different variable, an extra term arises--here, dy/dx.
So from the chain and product rules, d/dx (y^2) = 2y*dy/dx = 2y y'. Then the rest is straightforward:
y^2+y = x^3+2x
2y y' + y' = 3x^2 + 2
y'(1+2y) = 3x^2+2
y' = (3x^2+2)/(1+2y)
Sometimes the chain rule is put as df/dy*dy/dx = df/dx for making it easy to remember. (In my opinion the notation is unfortunate because it causes people to ignore the proof, and maybe think that differential elements behave like fractions, but I'm in the vast minority I'm sure.) Here, d(y^2)/dx = d(y^2)/dy*dy/dx = 2y y'.
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