Quotient Rule

Let y= f(x) g(x)  , then

dy dx = lim h0 f(x+h) g(x+h) f(x) g(x) h

= lim h0 f(x+h)g(x)f(x)g(x+h) g(x+h)g(x) h

= lim h0 f(x+h)g(x)f(x)g(x+h) hg(x)g(x+h)

= lim h0 f(x+h)g(x)f(x)g(x)f(x)g(x+h)+f(x)g(x) hg(x)g(x+h)

= lim h0 f(x+h)g(x)f(x)g(x) hg(x)g(x+h) f(x)g(x+h)f(x)g(x) hg(x)g(x+h)

= lim h0 [f(x+h)f(x)]g(x) hg(x)g(x+h) f(x)[g(x+h)g(x)] hg(x)g(x+h)

= lim h0 f(x+h)f(x) h g(x) g(x)g(x+h) lim h0 f(x) g(x)g(x+h) g(x+h)g(x) h

= lim h0 f(x+h)f(x) h lim h0 g(x) g(x)g(x+h) lim h0 f(x) g(x)g(x+h) lim h0 g(x+h)g(x) h

= f (x) g(x) [g(x)] 2 f(x) [g(x)] 2 g (x)

= f (x)g(x)f(x) g (x) [g(x)] 2