Prove the Quotient Rule

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Lesson Parent: 
Differentiation IV: Differentiation Formulas Everyone Must Know
Problem: 

What is the derivative of the quotient of $f(x)$ and $g(x)$ when both are differentiable and $g(x)\neq 0$? i.e. what is
\[\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right]?\]

Answer: 

\[\frac{g(x)\,f'(x)-f(x)\,g'(x)}{g(x)^2}\]

SAMPLE SOLUTION

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Solution: 

Let <Sign in to see all the formulas>. Now

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Now recall these three properties of functions limits,

  1. the limit of a constant times a function is the constant times the function's limit,
  2. the limit of the sum of two functions is the sum of the function's limits,
  3. and that the limit of the product of two functions is the product of the function's limits.

Note that $f(x)$ and $g(x)$ are constants with respect to the limit because they are not functions of $h$. The two functions

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are just $f'(x)$ and $g'(x)$ in the limit as <Sign in to see all the formulas> by the assumptions of the problem. Finally, we know that because $g(x)$ is differentiable it is continuous and that if it is continuous then <Sign in to see all the formulas> is continuous so that

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Therefore, all the limits in

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are well defined so that we can use our three bullet points above to state that

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