Differentiation of functions defined implicitly (C3)

Could somebody please explain (thoroughly) to me how to differentiate implicit functions e.g. y^4 + 2xy = x^2? My teacher has not been through this and shows no sign of doing so any time soon and I just can't understand how to do it!

Thanks in advance.

Reply 1

CubeDude

Take

\frac{d}{dx}

of each term, paying close attention to when you have to apply it to a term containing y, and when you might need to use the product rule.

Hint:

\frac{d}{dx} = \frac{d}{dy} \frac{dy}{dx}

Reply 2

bballer4life

i think basically differentiate it as you would as if they are all x terms, but when ever you differentiate a y, multiply the derivative by (dy/dx)

i know it's not very thorough, but still, you'd get

4y^3(dy/dx) + (using the product rule) [2y + 2x(dy/dx)] = 2x

[4y^3 + 2x](dy/dx) = 2x - 2y

(dy/dx) = [2x - 2y]/[4y^3 +2x]

Reply 3

Necro Defain

Indeed, in general, d/dx(y^n)=d/dy(y^n)dy/dx (chain rule) = ny^(n-1) dy/dx, and if you have multiples of powers of y and x, you use the product rule. To find dy/dx you then take it out as a factor of all your terms and re-arrange it.

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