# Calculus Examples

Find the first derivative of the function.
By the Sum Rule, the derivative of with respect to is .
Evaluate .
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 .
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.
By the Sum Rule, the derivative of with respect to is .
Evaluate .
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.
Divide each term in by .
Simplify the left side of the equation by cancelling the common factors.
Reduce the expression by cancelling the common factors.
Factor out of .
Move the negative one from the denominator of .
Simplify the expression.
Multiply by .
Rewrite as .
Simplify the right side of the equation.
Reduce the expression by cancelling the common factors.
Factor out of .
Cancel the common factors.
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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