# How do you find the intervals of increasing and decreasing using the first derivative given #y=1/(x+1)^2#?

Applying the Power Rule and Chain Rule

graph{1/(x+1)^2 [-3.847, 1.626, -0.537, 2.202]}

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To find the intervals of increasing and decreasing for the function ( y = \frac{1}{(x+1)^2} ) using the first derivative:

- Find the first derivative of the function ( y ) with respect to ( x ).
- Set the first derivative equal to zero and solve for ( x ) to find critical points.
- Determine the intervals where the first derivative is positive or negative to identify increasing and decreasing intervals.

Now, let's proceed with the steps:

- The first derivative of ( y ) with respect to ( x ) can be found using the chain rule:

[ \frac{d}{dx}\left(\frac{1}{(x+1)^2}\right) = -\frac{2}{(x+1)^3} ]

- Set the first derivative equal to zero and solve for ( x ) to find critical points:

[ -\frac{2}{(x+1)^3} = 0 ] [ (x+1)^3 \neq 0 ]

The expression ( (x+1)^3 ) is never zero, so there are no critical points.

- Determine the intervals where the first derivative is positive or negative:

The first derivative ( -\frac{2}{(x+1)^3} ) is negative for all ( x ) values except ( x = -1 ). At ( x = -1 ), the first derivative is undefined. Therefore, the function is decreasing for all ( x ) except at ( x = -1 ), where it's undefined.

So, the interval of decreasing is ( (-\infty, -1) \cup (-1, \infty) ). There are no intervals of increasing for this function.

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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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