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Help: derivatives

Moderator's edit:

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

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

Note by Akhilesh Prasad
1 year, 8 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 \((2)\): \[y'=(\dfrac{2x-y}{\not x})(\not x(x-y))\] \[=2x^2-3xy+y^2\]

Rishabh Cool - 1 year, 8 months ago

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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.

Akhilesh Prasad - 1 year, 8 months ago

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Thanks a lot.

Akhilesh Prasad - 1 year, 8 months ago

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