Proof of Quotient Rule

Key Questions

  • 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.

Questions