14 - Product 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+h)f(x)g(x+h)+f(x)g(x+h)f(x)g(x) h

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

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

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

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

= f (x)g(x)+f(x) g (x)