Question

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

Answer 1

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

Explanation:

Find the derivative of each part.

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

`d/dx(x)=1`

Use the chain rule:

`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`