# How do you use implicit differentiation to find the slope of the curve given #1/(x+1)+1/(y+1)=1# at (1,1)?

The curve has a slope of

The slope, or rate of change of the curve is

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then

NOTE. What is de difference between

Attached two graphics. The first is associated to

and the second is associated to

This second graphic is clean, without the singularities in

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To find the slope of the curve at the point (1,1) using implicit differentiation, differentiate both sides of the equation with respect to x, then solve for dy/dx. The equation is: ( \frac{1}{x+1} + \frac{1}{y+1} = 1 ). Differentiate both sides with respect to x: ( -\frac{1}{{(x+1)}^2} + \frac{1}{{(y+1)}^2} \cdot \frac{dy}{dx} = 0 ). Substitute x = 1 and y = 1 into the equation and solve for ( \frac{dy}{dx} ). This yields the slope of the curve at the point (1,1).

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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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