How can you derive the quotient rule?

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How can you derive the quotient rule?

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Answer 1 · 2014-08-19T14:33:51

This can be proven fairly quickly, assuming knowledge of prior subjects such as the product rule and chain rule. Suppose $f (x) = \frac{u (x)}{v (x)}$ $f(x) = (u(x))/(v(x))$ . As we know that all of our equations are in terms of $x$ $x$ , henceforth $x$ $x$ will be omitted from the steps below. Note however that it is still present as the variable for the functions.

$(\frac{d}{d x}) f = (\frac{d}{d x}) \frac{u}{v}$ $(d/dx)f = (d/dx)u/v$

Then via our definition $f = \frac{u}{v}$ $f= u/v$ we get $u = f \cdot v$ $u= f*v$ . Differentiating this via use of the product rule nets us...

$u' = f'*v + f*v'$

Now as we isolate f' on its own side...

$f'= [u'-f*v']/(v)$

Recalling that $f=u/v$ this becomes...

$f' = [u' - (u/v)*v']/v$

And by multiplying both the numerator and denominator by $v$ we get...

$f' = [u'*v - u*v']/[v^2]$

Or, by showing $x$ again...

$f'(x) = [u'(x)*v(x) - u(x)*v'(x)]/(v(x))^2$

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How can you derive the quotient rule?

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