# 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 .

Differentiate using the Constant Rule.

Since is constant with respect to , the derivative of with respect to is .

Add and .

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 .

Differentiate using the Constant Rule.

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.

Add to both sides of the equation.

Divide each term in by .

Cancel the common factor of .

Cancel the common factor.

Divide by .

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

is a local minimum

Replace the variable with in the expression.

Simplify the result.

Simplify each term.

Apply the product rule to .

Raise to the power of .

Raise to the power of .

Cancel the common factor of .

Factor out of .

Cancel the common factor.

Rewrite the expression.

Multiply .

Combine and .

Multiply by .

Move the negative in front of the fraction.

Find the common denominator.

Multiply by .

Multiply and .

Write as a fraction with denominator .

Multiply by .

Multiply and .

Reorder the factors of .

Multiply by .

Combine fractions.

Combine fractions with similar denominators.

Multiply by .

Simplify the numerator.

Subtract from .

Add and .

The final answer is .

These are the local extrema for .

is a local minima