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1 Product and Quotient Rules Product and Quotient Rules Goal To find the derivative of y D f .x/g.x/ from df dg and dx dx � f .x/g.x/ by separating f and g That same y is f .x C x/ Œg.x C x/ � g.x/ C g.x/ Œf .x C x/ � f .x/ Idea Write y D f .x C x/ g.x C x/ y g f D f .x C x/ C g.x/ x x x Product Rule dy dg df D f .x/ C g.x/ dx dx dx dy D x 2 cos x C 2x sin x dx A picture shows the two unshaded pieces of y D f .x C x/g C g.x/f Example y D x 2 sin x Product Rule g top area D .f .x/ C f /g g.x/ side area D g.x/f f .x/ Example f .x/ D x n Product Rule f g.x/ D x y D f .x/g.x/ D x nC1 dy dx dx n D xn Cx D x n C x nx n�1 D .n C 1/x n dx dx dx The correct derivative of x n leads to the correct derivative of x nC1 f .x/ dy df Quotient Rule If y D then D g.x/ g.x/ dx dx d EXAMPLE dx sin x cos x � f .x/ dg dx D .cos x.cos x/ � sin x.� sin x// g2 cos 2 x d 1 (Notice .cos x/2 C .sin x/2 D 1) tan x D D sec2 x dx cos 2 x d 1 x 4 times 0 � 1 times 4x 3 �4 EXAMPLE D D 5 This is nx n�1 dx x 4 x8 x f .x C x/ f .x/ f C f f � �g Prove the Quotient Rule y D D g.x C x/ g.x/ g C g This says that Write this y as g.f C f / � f .g C g/ gf � f g D g.g C g/ g.g C g/ Now divide that y by x As x � 0 we have the Quotient Rule 2 Product and Quotient Rules Practice Questions 1. Product Rule: Find the derivative of y D .x 3 /.x 4 /: Simplify and explain. 2. Product Rule: Find the derivative of y D .x 2 /.x �2 /: Simplify and explain. 3. Quotient Rule: Find the derivative of y D 4. Quotient Rule: Show that y D cos x : sin x sin x has a maximum (zero slope) at x D 0: x 5. Product and Quotient! Find the derivative of y D 6. g.x/ has a minimum when yD dg d 2g D 0 and dx dx 2 �0 x sin x : cos x The graph is bending up 1 dy d 2 y has a maximum at that point: Show that D 0 and g.x/ dx dx 2 �0 MIT OpenCourseWare http://ocw.mit.edu Resource: Highlights of Calculus Gilbert Strang The following may not correspond to a particular course on MIT OpenCourseWare, but has been provided by the author as an individual learning resource. For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
