Calculus Examples

Find Where Increasing/Decreasing Using Derivatives f(x)=x/(x^2+1)
Step 1
Find the first derivative.
Tap for more steps...
Step 1.1
Find the first derivative.
Tap for more steps...
Step 1.1.1
Differentiate using the Quotient Rule which states that is where and .
Step 1.1.2
Differentiate.
Tap for more steps...
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 the expression.
Tap for more steps...
Step 1.1.2.6.1
Add and .
Step 1.1.2.6.2
Multiply by .
Step 1.1.3
Raise to the power of .
Step 1.1.4
Raise to the power of .
Step 1.1.5
Use the power rule to combine exponents.
Step 1.1.6
Add and .
Step 1.1.7
Subtract from .
Step 1.2
The first derivative of with respect to is .
Step 2
Set the first derivative equal to then solve the equation .
Tap for more steps...
Step 2.1
Set the first derivative equal to .
Step 2.2
Set the numerator equal to zero.
Step 2.3
Solve the equation for .
Tap for more steps...
Step 2.3.1
Subtract from both sides of the equation.
Step 2.3.2
Divide each term in by and simplify.
Tap for more steps...
Step 2.3.2.1
Divide each term in by .
Step 2.3.2.2
Simplify the left side.
Tap for more steps...
Step 2.3.2.2.1
Dividing two negative values results in a positive value.
Step 2.3.2.2.2
Divide by .
Step 2.3.2.3
Simplify the right side.
Tap for more steps...
Step 2.3.2.3.1
Divide by .
Step 2.3.3
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 2.3.4
Any root of is .
Step 2.3.5
The complete solution is the result of both the positive and negative portions of the solution.
Tap for more steps...
Step 2.3.5.1
First, use the positive value of the to find the first solution.
Step 2.3.5.2
Next, use the negative value of the to find the second solution.
Step 2.3.5.3
The complete solution is the result of both the positive and negative portions of the solution.
Step 3
The values which make the derivative equal to are .
Step 4
Split into separate intervals around the values that make the derivative or undefined.
Step 5
Substitute a value from the interval into the derivative to determine if the function is increasing or decreasing.
Tap for more steps...
Step 5.1
Replace the variable with in the expression.
Step 5.2
Simplify the result.
Tap for more steps...
Step 5.2.1
Simplify the numerator.
Tap for more steps...
Step 5.2.1.1
Raise to the power of .
Step 5.2.1.2
Multiply by .
Step 5.2.1.3
Add and .
Step 5.2.2
Simplify the denominator.
Tap for more steps...
Step 5.2.2.1
Raise to the power of .
Step 5.2.2.2
Add and .
Step 5.2.2.3
Raise to the power of .
Step 5.2.3
Move the negative in front of the fraction.
Step 5.2.4
The final answer is .
Step 5.3
At the derivative is . Since this is negative, the function is decreasing on .
Decreasing on since
Decreasing on since
Step 6
Substitute a value from the interval into the derivative to determine if the function is increasing or decreasing.
Tap for more steps...
Step 6.1
Replace the variable with in the expression.
Step 6.2
Simplify the result.
Tap for more steps...
Step 6.2.1
Simplify the numerator.
Tap for more steps...
Step 6.2.1.1
Raising to any positive power yields .
Step 6.2.1.2
Multiply by .
Step 6.2.1.3
Add and .
Step 6.2.2
Simplify the denominator.
Tap for more steps...
Step 6.2.2.1
Raising to any positive power yields .
Step 6.2.2.2
Add and .
Step 6.2.2.3
One to any power is one.
Step 6.2.3
Divide by .
Step 6.2.4
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.
Tap for more steps...
Step 7.1
Replace the variable with in the expression.
Step 7.2
Simplify the result.
Tap for more steps...
Step 7.2.1
Simplify the numerator.
Tap for more steps...
Step 7.2.1.1
Raise to the power of .
Step 7.2.1.2
Multiply by .
Step 7.2.1.3
Add and .
Step 7.2.2
Simplify the denominator.
Tap for more steps...
Step 7.2.2.1
Raise to the power of .
Step 7.2.2.2
Add and .
Step 7.2.2.3
Raise to the power of .
Step 7.2.3
Move the negative in front of the fraction.
Step 7.2.4
The final answer is .
Step 7.3
At the derivative is . Since this is negative, the function is decreasing on .
Decreasing on since
Decreasing on since
Step 8
List the intervals on which the function is increasing and decreasing.
Increasing on:
Decreasing on:
Step 9