Calculus Examples

Find Where Increasing/Decreasing Using Derivatives
Step 1
Find the first derivative.
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Find the first derivative.
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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 .
Add and .
The first derivative of with respect to is .
Step 2
Set the first derivative equal to then solve the equation .
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Set the first derivative equal to .
Divide each term in by and simplify.
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Divide each term in by .
Simplify the left side.
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Cancel the common factor of .
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Cancel the common factor.
Divide by .
Simplify the right side.
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Divide by .
Take the cube root of both sides of the equation to eliminate the exponent on the left side.
Simplify .
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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.
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Replace the variable with in the expression.
Simplify the result.
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Raise to the power of .
Multiply by .
The final answer is .
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.
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Replace the variable with in the expression.
Simplify the result.
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One to any power is one.
Multiply by .
The final answer is .
At the derivative is . Since this is positive, the function is increasing on .
Increasing on since
Increasing on since
Step 7
List the intervals on which the function is increasing and decreasing.
Increasing on:
Decreasing on:
Step 8
Enter YOUR Problem
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