# Help: derivatives

Moderator's edit:

Given that $$x = y + \cfrac 1{y + \cfrac1{y + \cfrac1{y + \cfrac1{y + _\ddots}}}}$$.

Show that $$\dfrac{dy}{dx} = 2x^2 + y^2 - 3xy$$.

2 years, 3 months ago

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$x=y+\dfrac 1x \implies 1=x(x-y)\cdots (2)\\ \implies x^2=xy+1\cdots (1)$ Differentiating $$(1)$$ $y'=(\dfrac{2x-y}{x})(1)$ Using $$(2)$$: $y'=(\dfrac{2x-y}{\not x})(\not x(x-y))$ $=2x^2-3xy+y^2$

- 2 years, 3 months ago

I think the minute i opened the note and saw the moderator's edit, that was when the answer came to me. Kudos to the moderator.

- 2 years, 3 months ago

Thanks a lot.

- 2 years, 3 months ago