Calculus Examples

Find Where Increasing/Decreasing Using Derivatives
Step 1
Find the first derivative.
Find the first derivative.
By the Sum Rule, the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Since is constant with respect to , the derivative of with respect to is .
The first derivative of with respect to is .
Step 2
Set the first derivative equal to then solve the equation .
Set the first derivative equal to .
Divide each term in by and simplify.
Divide each term in by .
Simplify the left side.
Cancel the common factor of .
Cancel the common factor.
Divide by .
Simplify the right side.
Divide by .
Take the cube root of both sides of the equation to eliminate the exponent on the left side.
Simplify .
Rewrite as .
Pull terms out from under the radical, assuming real numbers.
Step 3
The values which make the derivative equal to are .
Step 4
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 5
Substitute a value from the interval into the derivative to determine if the function is increasing or decreasing.
Replace the variable with in the expression.
Simplify the result.
Raise to the power of .
Multiply by .
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.
Replace the variable with in the expression.
Simplify the result.
One to any power is one.
Multiply by .