Calculus Examples

Find Where Increasing/Decreasing Using Derivatives
Find the 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 .
Set the derivative equal to .
Solve for .
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Divide each term by and simplify.
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Divide each term in by .
Cancel the common factor of .
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Cancel the common factor.
Divide by .
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.
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 .
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
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
List the intervals on which the function is increasing and decreasing.
Increasing on:
Decreasing on:
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