I understand the product rule and how to use it, but I've been told to prove it, and I don't really know how. Could I be helped with this?
Thank you
There's only a couple of steps left to the proof, I don't want to spoil it completely!
$\displaystyle f'(x) = \lim_{h \to 0}\frac{v(x+h)u(x+h)-v(x)u(x)}{h}$
here's a little bit more
$\displaystyle f'(x) = \lim_{h \to 0}\frac{v(x+h)u(x+h)-v(x+h)u(x)+v(x+h)u(x)-v(x)u(x)}{h}$
Now factor out $\displaystyle v(x+h)$ and $\displaystyle u(x)$ in groups.
I may have got it I think,
$\displaystyle \frac{v(x+h)u(x+h)-v(x+h)u(x)+v(x+h)u(x)-v(x)u(x)}{h}$
$\displaystyle = v(x+h)\frac{u(x+h)-u(x)}{h}+u(x)\frac{v(x+h)-v(x)}{h}$
but $\displaystyle \frac{u(x+h)-u(x)}{h}$ can be written as $\displaystyle u'(x)$ as h tends to zero
thus $\displaystyle f'(x)=v(x)u'(x)+u(x)v'(x)$ as h tends to zero
am I correct?