Calculus Examples

Find Where Increasing/Decreasing Using Derivatives f(x)=x/(x+1)
Step 1
Find the first derivative.
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Step 1.1
Find the first derivative.
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Step 1.1.1
Differentiate using the Quotient Rule which states that is where and .
Step 1.1.2
Differentiate.
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Step 1.1.2.1
Differentiate using the Power Rule which states that is where .
Step 1.1.2.2
Multiply by .
Step 1.1.2.3
By the Sum Rule, the derivative of with respect to is .
Step 1.1.2.4
Differentiate using the Power Rule which states that is where .
Step 1.1.2.5
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.2.6
Simplify by adding terms.
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Step 1.1.2.6.1
Add and .
Step 1.1.2.6.2
Multiply by .
Step 1.1.2.6.3
Subtract from .
Step 1.1.2.6.4
Add and .
Step 1.2
The first derivative of with respect to is .
Step 2
Set the first derivative equal to then solve the equation .
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Step 2.1
Set the first derivative equal to .
Step 2.2
Set the numerator equal to zero.
Step 2.3
Since , there are no solutions.
No solution
No solution
Step 3
There are no values of in the domain of the original problem where the derivative is or undefined.
No critical points found
Step 4
Find where the derivative is undefined.
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Step 4.1
Set the denominator in equal to to find where the expression is undefined.
Step 4.2
Solve for .
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Step 4.2.1
Set the equal to .
Step 4.2.2
Subtract from both sides of the equation.
Step 5
After finding the point that makes the derivative equal to or undefined, the interval to check where is increasing and where it is decreasing is .
Step 6
Substitute a value from the interval into the derivative to determine if the function is increasing or decreasing.
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Step 6.1
Replace the variable with in the expression.
Step 6.2
Simplify the result.
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Step 6.2.1
Simplify the denominator.
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Step 6.2.1.1
Add and .
Step 6.2.1.2
Raise to the power of .
Step 6.2.2
Divide by .
Step 6.2.3
The final answer is .
Step 6.3
At the derivative is . Since this is positive, the function is increasing on .
Increasing on since
Increasing on since
Step 7
Substitute a value from the interval into the derivative to determine if the function is increasing or decreasing.
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Step 7.1
Replace the variable with in the expression.
Step 7.2
Simplify the result.
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Step 7.2.1
Simplify the denominator.
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Step 7.2.1.1
Add and .
Step 7.2.1.2
One to any power is one.
Step 7.2.2
Divide by .
Step 7.2.3
The final answer is .
Step 7.3
At the derivative is . Since this is positive, the function is increasing on .
Increasing on since
Increasing on since
Step 8
List the intervals on which the function is increasing and decreasing.
Increasing on:
Step 9