Proof of Quotient Rule

Key Questions

  • How do you prove the quotient rule?

    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 · 1 · Oct 2 2014

Questions

Basic Differentiation Rules