Where on the x-axis is $f(x)=1/(x+1)$ increasing or decreasing?

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Lesson Parent: 
Geometry of Functions IV: Using Derivatives To Identify Increasing and Decreasing Functions
Problem: 

Where on the x-axis is $f(x)=1/(x+1)$ increasing or decreasing?

Answer: 

$f(x)=1/(x+1)$ decreases on both $(-\infty,-1)$ and $(-1,\infty)$.

SAMPLE SOLUTION

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Solution: 

First, note that the function is not defined at $x=-1$. Therefore, the function is defined on the intervals <Sign in to see all the formulas> and <Sign in to see all the formulas>. We can also graph the function to see that the function is probably decreasing on both of these intervals.

Graph of f(x)=1/(x+1)

But now let's use the function's derivative to prove it.

When <Sign in to see all the formulas> we can use the quotient rule to find

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When <Sign in to see all the formulas> then <Sign in to see all the formulas>. When <Sign in to see all the formulas> then <Sign in to see all the formulas> and therefore <Sign in to see all the formulas>. Similarly, when <Sign in to see all the formulas> then <Sign in to see all the formulas>. When <Sign in to see all the formulas> then <Sign in to see all the formulas> and therefore <Sign in to see all the formulas>.

Since

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for both intervals it is decreasing on both intervals.