Proof of the Quotient Rule

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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

Proof By the definition of the derivative,
dx 
g 
(x)  = 


= 

(subtraction of fractions)  
= 

(we added & subtracted f(x)g(x))  
= 

(a little algebra)  
= 

(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
dx 
g 
(x)  =  [g(x)]^{2}  , 
which is the quotient rule.