Calculus Examples

Find Where Increasing/Decreasing Using Derivatives
Find the derivative.
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Differentiate.
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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 .
Evaluate .
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Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Multiply by .
Differentiate using the Constant Rule.
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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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Factor out of .
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Factor out of .
Factor out of .
Factor out of .
Divide each term by and simplify.
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Divide each term in by .
Reduce the expression by cancelling the common factors.
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Cancel the common factor.
Divide by .
Divide by .
Set equal to and solve for .
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Set the factor equal to .
Add to both sides of the equation.
Take the square root of both sides of the equation to eliminate the exponent on the left side.
The complete solution is the result of both the positive and negative portions of the solution.
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First, use the positive value of the to find the first solution.
Next, use the negative value of the to find the second solution.
The complete solution is the result of both the positive and negative portions of the solution.
The solution is the result of and .
The values which make the derivative equal to are .
Split into separate intervals around the values that make the derivative or undefined.
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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Simplify each term.
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Raise to the power of .
Multiply by .
Multiply by .
Add and .
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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Simplify each term.
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Raise to the power of .
Multiply by .
Multiply by .
Add and .
The final answer is .
At the derivative is . Since this is positive, the function is increasing on .
Increasing on since
Increasing 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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Simplify each term.
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Raise to the power of .
Multiply by .
Multiply by .
Subtract from .
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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Simplify each term.
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Raise to the power of .
Multiply by .
Multiply by .
Subtract from .
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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