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

Question

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

Samantha Adkison · Accepted Answer

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

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#

Christopher Allers · Answer

undefined To implicitly differentiate the equation -y = x - sqrt(x - y) with respect to x:

1. Differentiate each term with respect to x.
2. Treat y as a function of x and apply the chain rule when differentiating terms containing y.
3. Solve for dy/dx.

Differentiating each term:
- Differentiate -y with respect to x: -dy/dx.
- Differentiate x with respect to x: 1.
- Differentiate sqrt(x - y) with respect to x: (1/2) * (x - y)^(-1/2) * (1 - dy/dx).

Putting it all together:
-(-dy/dx) = 1 - (1/2) * (x - y)^(-1/2) * (1 - dy/dx).

Solve for dy/dx:
dy/dx = 1 - (1/2) * (x - y)^(-1/2) * (1 - dy/dx).

Multiply through by 2:
2dy/dx = 2 - (x - y)^(-1/2) * (1 - dy/dx).

Add dy/dx to both sides:
3dy/dx = 2 - (x - y)^(-1/2).

Finally, isolate dy/dx:
dy/dx = (2 - (x - y)^(-1/2)) / 3.