Proof of the Quotient Rule
to accompany
Calculus Applied to the Real World

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The Quotient Rule

If the functions f and g are differentiable at x, with g(x) 0, then the quotient f/g is differentiable at x, and

    d

    dx
    f

    g
    (x) =
    f'(x) g(x) - f(x) g'(x)

    [g(x)]2
    .

 
Proof By the definition of the derivative,

If we recognize the difference quotients for f and g in this last expression, we see that taking the limit as h0 replaces them by the dreivatives f'(x) and g'(x). Further, since g is differentiable, it is also continuous, and so g(x+h)g(x) as h0. Putting this all together gives

which is the quotient rule.