Proof of Quotient Rule
Key Questions
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By the definition of the derivative,
#[{f(x)}/{g(x)}]'=lim_{h to 0}{f(x+h)/g(x+h)-f(x)/g(x)}/{h}# by taking the common denominator,
#=lim_{h to 0}{{f(x+h)g(x)-f(x)g(x+h)}/{g(x+h)g(x)}}/h# by switching the order of divisions,
#=lim_{h to 0}{{f(x+h)g(x)-f(x)g(x+h)}/h}/{g(x+h)g(x)}# by subtracting and adding
#f(x)g(x)# in the numerator,#=lim_{h to 0}{{f(x+h)g(x)-f(x)g(x)-f(x)g(x+h)+f(x)g(x)}/h}/{g(x+h)g(x)}# by factoring
#g(x)# out of the first two terms and#-f(x)# out of the last two terms,#=lim_{h to 0}{{f(x+h)-f(x)}/h g(x)-f(x){g(x+h)-g(x)}/h}/{g(x+h)g(x)}# by the definitions of
#f'(x)# and#g'(x)# ,#={f'(x)g(x)-f(x)g'(x)}/{[g(x)]^2}# I hope that this was helpful.
Wataru
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Oct 2 2014