Question

How can you derive the quotient rule?

Answer 1

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

`(d/dx)f = (d/dx)u/v`

Then via our definition `f= u/v` we get `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`