Question

Find the implicit differentiation for each.

1) x^2 = (x-y) / (x+y)

2) (x+y)^3 + (x-y)^3 = x^4 + y^4

Answer 1

Just keep in mind the chain rule. Every time you have a y, you will produce a y'.

x^2 = (x-y)/(x+y)
2x = ((1-y')(x+y) - (x-y)(1+y'))/(x+y)^2

At this point, it's just algebra. Collect all the y' stuff on one side, and you wind up with (one form, anyway):

y' = -(x^3 + 2x^2y + xy^2 - y)/x

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(x+y)^3 + (x-y)^3 = x^4 + y^4
3(x+y)^2 (1+y') + 3(x-y)^2 (1-y') = 4x^3 + 4y^3 y'

y' =

(3-2x)x^2 + 3y^2
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2(y^3-3xy)