Question

$implicit\:derivative\:\frac{dy}{dx},\:\left(x-y\right)^2=x+y-1$

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Answer 1

$\mathrm{Implicit\:Derivative\:}\frac{dy}{dx}\mathrm{\:of\:}\left(x-y\right)^2=x+y-1:\quad \frac{1-2x+2y}{-2x+2y-1}$

$\left(x-y\right)^2=x+y-1$

$\mathrm{Treat\:}y\mathrm{\:as\:}y\left(x\right)$

$\mathrm{Differentiate\:both\:sides}:\quad 2\left(x-y\right)\left(1-\frac{d}{dx}\left(y\right)\right)=\frac{d}{dx}\left(y\right)+1$

$2\left(x-y\right)\left(1-\frac{d}{dx}\left(y\right)\right)=\frac{d}{dx}\left(y\right)+1$

$\mathrm{Isolate}\:\frac{d}{dx}\left(y\right):\quad \frac{d}{dx}\left(y\right)=\frac{1-2x+2y}{-2x+2y-1}$

$\frac{d}{dx}\left(y\right)=\frac{1-2x+2y}{-2x+2y-1}$

Answer 2

• To Implicitly derive a function (useful when a function can't easily be solved for y) Differentiate with respect to x. Collect all the dy/dx on one side. Solve for dy/dx.

• To derive an inverse function, restate it without the inverse then use Implicit differentiation.

$\mathrm{Implicit\:Derivative\:}\frac{dy}{dx}\mathrm{\:of\:}\left(x-y\right)^2=x+y-1:\quad \frac{1-2x+2y}{-2x+2y-1}$