Derive the quotient rule from the definition of a differential?

Question

Heres another one. I know how to derive when y=f(x) but im struggling with this one. thanks.

Answer 1

d/dx[f(x)/g(x)] =
lim h->0 [f(x+h)/g(x+h) - f(x)/g(x)] / h =
lim h->0 [f(x+h)g(x) - f(x)g(x+h)] / [hg(x)g(x+h)] =
lim h->0 [f(x+h)g(x) - f(x)g(x) + f(x)g(x) - f(x)g(x+h)]/[hg(x)g(x+h)] =
lim h->0 [(f(x+h)-f(x))g(x) - f(x)(g(x+h)-g(x))]/[hg(x)g(x+h)] =
lim h->0 [g(x)(f(x+h)-f(x))/h -f(x)(g(x+h)-g(x))/h]/[g(x)g(x+h)] =
[g(x)f'(x)-f(x)g'(x)]/[g(x)^2]

Note that this requires f(x) and g(x) to be differentiable, of course, so that their derivatives exist. Also, since g(x) is differentiable, it's also continuous, which is why you know that lim h->0 g(x+h) = g(x).

Answer 2

Watch your language and keep your quotients to yourself!

Answer 3

HA!