The Product and Quotient Rules and Higher-Order Derivatives Calculus 2.3

Product Rule Take the first function times the derivative of the second plus the second function times the derivative of the first.

Product rule with more than two functions Take the derivative of each function times the other functions and add all products

Examples Find the derivatives using the product rule.

The quotient rule Take the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator divided by the square of the denominator.

The quotient rule

Examples Find the derivatives using the quotient rule

Avoiding the quotient rule See example 6 on page 130

Derivatives of trig functions Can be proven using the quotient rule

Higher order derivatives Second derivative of the function is the derivative of the first derivative Third derivative of the function is the derivative of the second derivative Etc. See page 133

Acceleration The derivative of velocity. The second derivative of position. Example: find the acceleration of an object in free fall if the position function is