Question

Quotient Rule Derivation Using Product Rule

Answer 1

Quotient Rule basically says that:

\[\large{\cfrac{d}{dx}(\cfrac{u(x)}{v(x)}) = \cfrac{u'(x)v(x) - v'(x)u(x)}{(v(x))^2}}\]

Answer 2

Now product rule states that:

\[\large{(fg)' = f'g + gf'}\]

Now let:

f(x) = u(x) and \(\large{g(x) = \cfrac{1}{v(x)}}\)

Then, using product rule we have:

\[\large{(fg)' = f'g + fg'}\]

Now,

\[\large{f'(x) = u'(x)}\]

and

\[\large{g'(x) = -\cfrac{1}{(v(x))^2}\cdot v'(x)}\]

Answer 3

Thus we have:

\[\large{(fg)' = f'g + fg'}\]

\[\large{\implies (fg)' = -u(x)\cdot\cfrac{v'(x)}{(v(x))^2} + u'(x)\cdot \cfrac{1}{v(x)}}\]

\[\large{\implies (u/v)' = \cfrac{vu' - uv'}{v^2}}\]

Well pretty easy and basic. Similarly, we can find out the similarity between different formulae which are basically the same.