2.3 Product & Quotient Rules; Higher-Order DerivativesMAC2233 MDC-North2.3 Product & Quotient Rules; Higher-Order DerivativesProduct Rule: If f(x) and g(x) are differentiable at x, then so is their product P(x) =f(x) g(x) and ( ) ( ) ( ) ( ) ( ) ( )d d df x g x f x g x g x f xdx dx dx� � � � � �= +� � � � � � or (fg)’ = fg’ + gf ’; the derivative of theproduct fg is f times the derivative of g plus g times the derivative of f.Quotient Rule: If f(x) and g(x) are differentiable functions, then so is their quotient Q(x) =f(x)/ g(x) and ( )( )( ) ( ) ( ) ( )( )( ),d dg x f x f x g xf xddx dxg xdx g x g x� � � �-� � � �� �= �� �� �20 or ' ''f gff gg g� �-=� �� �2. Second derivative is the derivative of the derivative. If y = f(x), the second derivativeis denoted by d ydx2 of f ‘’(x). The second derivative gives the rate of change of the rate of change of theoriginal function. The nth derivative For any positive integer n, the nth derivative of a function isobtained from the function by differentiating successively n times. If the originalfunction is y = f(x), the nth derivative is denoted by nd ydx or ( )nf x.Differentiate the given function; simplify:1.( ) ( ) ( )f x x x= + -7 5 3(product rule)2.( )( )( )f x x x x x=- - + +33 5 2 5 23.yx=+53 24.( )( )( )x xg xx x- +=+ +225 73 8MAC2233 MDC-North5.( )( )( )x xg xxx+ +=--21241Find an equation for the tangent line to the curve at the point when x=x0:6.;xy xx+= =-0705 27. A bacterial colony is estimated to have a population of ( )tP tt+=+224 101 million t hours after theintroduction of a toxin. [a] At what rate is the population changing one hour after the toxin isintroduced; is the population increasing or decreasing? [b] At what time does the populationbegin to decline?8. The owner of a boutique estimates that when a particular kind of perfume is priced at pdollars per bottle she will sell ( ),B p pp= �+50053, bottles a month at a total cost of C(p) =0.2p2 + 3p + 200. [a] Express the profit P(p) as a function of the price per bottle. [b] Atwhat rate is profit changing with respect to p when the price is $12 per bottle; is it increasingor
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