Calculus Examples

Find the first derivative.
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By the Sum Rule, the derivative of with respect to is .
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 .
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.
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Differentiate using the Power Rule which states that is where .
Since is constant with respect to , the derivative of with respect to is .
Simplify.
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Add and .
Reorder terms.
Graph each side of the equation. The solution is the x-value of the point of intersection.
Split into separate intervals around the values that make the first derivative or undefined.
Substitute any number, such as , from the interval in the first derivative to check if the result is negative or positive.
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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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Raising to any positive power yields .
Multiply by .
Raising to any positive power yields .
Multiply by .
Simplify by adding numbers.
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Add and .
Add and .
The final answer is .
Substitute any number, such as , from the interval in the first derivative to check if the result is negative or positive.
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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 .
Raise to the power of .
Multiply by .
Simplify by adding numbers.
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Add and .
Add and .
The final answer is .
Since the first derivative changed signs from positive to negative around , then there is a turning point at .
Find the y-coordinate of to find the turning point.
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Find to find the y-coordinate of .
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Replace the variable with in the expression.
Simplify .
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Remove parentheses.
Simplify each term.
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Raise to the power of .
Multiply by .
Raise to the power of .
Multiply by .
Simplify by adding and subtracting.
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Subtract from .
Add and .
Add and .
Write the and coordinates in point form.
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