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

Find the derivative of each part.

Use the chain rule:

Combine them all (sum rule):

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

- Differentiate each term with respect to x.
- Treat y as a function of x and apply the chain rule when differentiating terms containing y.
- 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.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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