Derivation of quotient rule of differentiation from first principles?

To derive the quotient rule of differentiation from first principles, we start with the definition of the derivative of a function f(x) at a point x=a:

f'(a) = lim(h->0) [f(a+h) - f(a)] / h

Now, let's consider a function g(x) = u(x) / v(x), where u(x) and v(x) are differentiable functions. We want to find the derivative of g(x) using the definition above.

Using the definition of the derivative, we have:

g'(a) = lim(h->0) [g(a+h) - g(a)] / h
= lim(h->0) [(u(a+h) / v(a+h)) - (u(a) / v(a))] / h
= lim(h->0) [(u(a+h)v(a) - u(a)v(a+h)) / (v(a+h)v(a))] / h
= lim(h->0) [(u(a+h)v(a) - u(a)v(a+h)) / h] / v(a+h)v(a)

Now, we can rewrite the expression inside the limit as:

= lim(h->0) [u(a+h)v(a) - u(a)v(a+h)] / h * 1 / v(a+h)v(a)
= lim(h->0) [u(a+h)v(a) - u(a)v(a+h)] / h * 1 / v(a) * 1 / v(a+h)

Now, we can apply the definition of the derivative to each term separately:

= [u'(a)v(a) + u(a)v'(a) - u(a)v(a)] / v(a)^2

Therefore, the derivative of the quotient of two functions u(x) and v(x) is given by:

(g(x))' = [u'(x)v(x) + u(x)v'(x) - u(x)v(x)] / v(x)^2

This is the quotient rule of differentiation, derived from first principles.