# Quotient rule for derivatives

Discussion in 'General Math' started by Ryan Stewart, Dec 23, 2004.

1. ### Ryan StewartGuest

I'm studying calculus independently, and an exercise in the book I'm using
is:
Derive the quotient rule. (Find y' for y = u(x)/v(x).)

The answer I came up with matches an intermediate step shown in the answers
section, but not the final answer. I think the book is wrong. My answer:
u'/v - (uv')/v^2

Process:
y = u/v = u(v^-1)
Product rule:
f(x) = u, g(x) = v^-1
f'(x) = u'
Chain rule to find g'(x):
Let g(x) = j(k(x)) = v^-1 and k(x) = w = v. Then j(w) = w^-1
j'(w) = -w^-2, k'(x) = v'
j'(x) = (-v^-2)v'
g'(x) = -v'/v^2
Finally:
y' = f(x)g'(x) + f'(x)g(x) = -uv'/v^2 + u'/v
y' = u'/v - uv'/v^2

The book shows this intermediate step which coincides with my answer:
y' = u(-1)(v^-2)v' + u'/v

But the final answer shown is this:
vu'/v^2 - uv'/v^2

Can anyone comment on this?

Ryan Stewart, Dec 23, 2004

2. ### N. SilverGuest

Find the lowest common denominator,
LCD = v^2.
What you have is the same as
vu'/v^2 - (uv')/v^2
= (vu'- uv')/v^2,

N. Silver, Dec 23, 2004