Section 2.3: Product and Quotient Rule

Objective: Students will be able to use the product and quotient rule to take the derivative of differentiable equations

Review: Definition of Derivative The derivative of f at x is given by Provided the limit exists. For all x for which this limit exists, f’ is a function of x.

Product Rule Theorem 2.7:The product of two differentiable function f and g is itself differentiable. Moreover, the derivative of fg is the first function times the derivative of the second, plus the second function times the derivative of the first. You can reverse the order in which you take the derivative of the terms in the product rule.

Example #1 (derivative of the second term)(first term)+(derivative of the first term)(second term) Step 1: Step 2: -Take derivative Step 3: -Simplify Step 4: -Simplify

Example #2 Step 1: -Take derivative Step 2: -Simplify Step 3: -Simplify

Example #3 Find the tangent line at point (-2,1) using the above equation -Plug slope & point into the point slope equation Step 6: -plug in x=-2 from the point (-2,1) to get the slope of the tangent line Step 4:Step 1: -Take derivative Step 2: -Simplify Step 3: -Simplify Step 5: -Simplify

Quotient Rule Theorem 2.8:The quotient f/g of two differentiable functions f and g is itself differentiable at all values of x for which g(x) 0. Moreover, the derivative of f/g is given by the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator. You can not reverse the order in which you take the derivative of the terms in the quotient rule. ≠

Example #1 Step 1: (derivative of the top term)(bottom term)-(derivative of the bottom term)(top term) (bottom term) 2 Step 2: -Take derivative Step 3: -Simplify

Example #2 Step 1: -Get rid of fraction in the numerator by multiply the numerator and denominator by x Step 2: -Simplify Step 3: -Take derivative Step 4: -Simplify

Example #3 Find tangent equation at point (-1,3) Step 1: -Take derivative Step 2: -Simplify Step 3: -Simplify Step 4: -plug in x=1 from the point (-1,3) to find the slope of the tangent line Step 5: -plug the slope and point into the point slope formula

Combining the Product Rule & Quotient Rule Step 2: -Simplify Step 3: -Simplify Step 4: -Simplify *For this type of problem use the quotient rule and with in the quotient rule use the product rule to take the derivative of the numerator Step 1: -Take derivative Product rule for derivative of the numerator