Proof of the Quotient Rule
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Techniques of Differentiation Topic Summary
Review Exercises on Techniques Index of Calculus Proofs Return to Main Page Everything for Calculus |
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The Quotient Rule
If the functions f and g are differentiable at x, with g(x)
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Proof By the definition of the derivative,
 dx |
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g |
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(x) | = |
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| = |
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(subtraction of fractions) | ||||||||||||
| = |
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(we added & subtracted f(x)g(x)) | ||||||||||||
| = |
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(a little algebra) | ||||||||||||
| = |
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(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 h
0 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 |
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g |
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(x) | = | [g(x)]2
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which is the quotient rule.
0, then the quotient f/g is differentiable at x, and