## 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

 ddx fg (x) = f'(x) g(x) - f(x) g'(x)[g(x)]2 .

Proof By the definition of the derivative,

d

dx
f

g
(x) =
lim
h0
 f(x+h)g(x+h) - f(x)g(x)

h
=  limh0 f(x+h)g(x) - f(x)g(x+h)g(x+h)g(x)h
(subtraction of fractions)
=  limh0 f(x+h)g(x) - f(x)g(x) + f(x)g(x) - f(x)g(x+h)g(x+h)g(x)h
=  limh0 [f(x+h) - f(x)]g(x) - f(x)[(g(x+h) - g(x)]g(x+h)g(x)h
(a little algebra)
=
lim
h0
 f(x+h)-f(x)h g(x) - f(x) g(x+h)-g(x)h

g(x+h)g(x)
(a little more algebra)

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

 ddx fg (x) = f'(x) g(x) - f(x) g'(x)[g(x)]2 ,

which is the quotient rule.