Calculus Examples

Find the first derivative of the function.
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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 .
Since is constant with respect to , the derivative of with respect to is .
Add and .
Find the second derivative of the function.
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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 .
Since is constant with respect to , the derivative of with respect to is .
Add and .
To find the local maximum and minimum values of the function, set the derivative equal to and solve.
Subtract from both sides of the equation.
Divide each term by and simplify.
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Divide each term in by .
Simplify the left side of the equation by cancelling the common factors.
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Reduce the expression by cancelling the common factors.
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Factor out of .
Move the negative one from the denominator of .
Simplify the expression.
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Multiply by .
Rewrite as .
Simplify the right side of the equation.
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Reduce the expression by cancelling the common factors.
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Factor out of .
Cancel the common factors.
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Move the negative in front of the fraction.
Evaluate the second derivative at . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.
is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.
is a local maximum
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