# Calculus Examples

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 .

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 .

To find the local maximum and minimum values of the function, set the derivative equal to and solve.

Factor out of .

Factor out of .

Factor out of .

Divide each term in by .

Cancel the common factor of .

Cancel the common factor.

Divide by .

Divide by .

Set the factor equal to .

Subtract from both sides of the equation.

The solution is the result of and .

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.

Multiply by .

Subtract from .

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

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.

Multiply by .

Subtract from .

is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

is a local minimum

These are the local extrema for .

is a local maximum

is a local minimum