# 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 .
Differentiate using the Constant Rule.
Since is constant with respect to , the derivative of with respect to is .
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 .
Differentiate using the Constant Rule.
Since is constant with respect to , the derivative of with respect to is .
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 by and simplify.
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
Find the y-value when .
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 .