How do you implicitly differentiate $- y = x - \sqrt{x - y}$ $-y=x-sqrt(x-y)$ ?

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Answer 1 · 2015-12-30T04:36:43

$\frac{d y}{d x} = \frac{1 - 2 \sqrt{x - y}}{1 + 2 \sqrt{x - y}}$ $dy/dx=(1-2sqrt(x-y))/(1+2sqrt(x-y))$

Find the derivative of each part.

$d/dx(-y)=-y'$

$d/dx(x)=1$

$d/dxsqrt(x-y)=1/(2sqrt(x-y))*(1-y')=(1-y')/(2sqrt(x-y))$

Combine them all (sum rule):

$-y'=1-(1-y')/(2sqrt(x-y))$

Multiply everything by $2sqrt(x-y)$ .

$-2y'sqrt(x-y)=2sqrt(x-y)-1+y'$

Isolate $y'$ .

$1-2sqrt(x-y)=y'+2y'sqrt(x-y)$

$y'(1+2sqrt(x-y))=1-2sqrt(x-y)$

$y'=(1-2sqrt(x-y))/(1+2sqrt(x-y))=dy/dx$

How do you implicitly differentiate -y=x-sqrt(x-y) −y=x−√x−y-y=x-sqrt(x-y) ?