Derivatives of Products and Quotients Michael Freeze MAT 151 UNC - PowerPoint PPT Presentation

Derivatives of Products and Quotients Michael Freeze MAT 151 UNC Wilmington Summer 2013 1 / 13

Product Rule Let f ( x ) = u ( x ) · v ( x ) where both u ′ ( x ) and v ′ ( x ) exist. Then f ′ ( x ) = u ( x ) · v ′ ( x ) + v ( x ) · u ′ ( x ) . That is, the derivative of a product of two functions is the first function times the derivative of the second plus the second function times the derivative of the first. 2 / 13

Product Rule Verification Product Rule d dx [ u ( x ) · v ( x )] = u ( x ) · v ′ ( x ) + v ( x ) · u ′ ( x ) Let f ( x ) = (3 x 2 + 2)(2 x − 1). (a) Express f ( x ) as a polynomial in standard form and find f ′ ( x ) by differentiating term-by-term. (b) Use the product rule to find f ′ ( x ) directly. 3 / 13

Product Rule Examples Product Rule d dx [ u ( x ) · v ( x )] = u ( x ) · v ′ ( x ) + v ( x ) · u ′ ( x ) Use the product rule to find the derivative of the function. f ( x ) = ( x − 5)(1 − 2 x ) 4 / 13

Product Rule Examples Product Rule d dx [ u ( x ) · v ( x )] = u ( x ) · v ′ ( x ) + v ( x ) · u ′ ( x ) Use the product rule to find the derivative of the function. y = ( x 5 − 2 x 3 ) x − 1 � � x 5 / 13

Product Rule Examples Product Rule d dx [ u ( x ) · v ( x )] = u ( x ) · v ′ ( x ) + v ( x ) · u ′ ( x ) Use the product rule to find the derivative of the function. g ( x ) = ( x + 1)( √ x + 2) 6 / 13

Quotient Rule Let f ( x ) = u ( x ) v ( x ) where both u ′ ( x ) and v ′ ( x ) exist and v ( x ) � = 0. Then f ′ ( x ) = v ( x ) · u ′ ( x ) − u ( x ) · v ′ ( x ) [ v ( x )] 2 . That is, the derivative of a quotient is 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. 7 / 13

Quotient Rule Verification Quotient Rule = v ( x ) · u ′ ( x ) − u ( x ) · v ′ ( x ) � u ( x ) � d [ v ( x )] 2 dx v ( x ) Let f ( x ) = 3 x 2 − 3 x +2 . x (a) Express f ( x ) as a sum of power functions and find f ′ ( x ) by differentiating term-by-term. (b) Use the quotient rule to find f ′ ( x ) directly. 8 / 13

Quotient Rule Examples Quotient Rule = v ( x ) · u ′ ( x ) − u ( x ) · v ′ ( x ) � u ( x ) � d [ v ( x )] 2 dx v ( x ) Use the quotient rule to find the derivative of the function. f ( x ) = 2 x − 5 3 x +1 9 / 13

Quotient Rule Examples Quotient Rule = v ( x ) · u ′ ( x ) − u ( x ) · v ′ ( x ) � u ( x ) � d [ v ( x )] 2 dx v ( x ) Use the quotient rule to find the derivative of the function. y = x 2 − 2 x +4 x − 1 10 / 13

Equation of Tangent Line Find an equation of the tangent line to the curve x f ( x ) = at x = 3 . x − 2 11 / 13

Average Cost The total cost (in hundreds of dollars) to produce x units of perfume is C ( x ) = 3 x + 2 x + 4 . Find the average cost for each production level. (a) 10 units (b) 20 units (c) x units (d) Find the marginal average cost function. 12 / 13

Memory Retention Some psychologists contend that the number of facts of a certain type that are remembered after t hours is given by 90 t f ( t ) = 99 t − 90 . Find the rate at which the number of facts remembered is changing after the following numbers of hours. (a) 1 (b) 10 13 / 13