MATH 140 Lecture Notes - Lecture 10: Quotient Rule

• School: University of Maryland
• Department: Mathematics
• Course: MATH 140
• Professor: Denny Gulick

Document Summary

Review d dx( 1 x)=: quotient rule: d dx( 1 x)= xn(0 ) (n xn 1) x2n. = n xn 1 x 2n= m x n 1=m xm 1: this implies that d dx (xk)=k xk 1 if k is any nonzero integer, ex: d dx ( x)=lim t x. The chain rule: definition: assume that f (a) g" and f " (a) exist. Then ( g f )" (x )=[g"(f (a))][ f "(a )] Idea of a proof: assume in addition that f (x) f (a) for x near a, thus, ( g f )( x) (g f )( a) x a g(fx) g( f (x)) g(f (a)) x a. = lim y f (a ) g ( y) g(f (a)) y f (a) Lim x a f (x) f (a) x a =[g"(f ( a))][f " (a)] Examples: ex1: k ( x)= 4 x2 k" ( x)=? k" ( x)=1.