Proof of the Product Rule

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

Shouldn't be too hard

Here's the start



Now by

we have



can you take it from here?

I have understood what you have done using the calculus theorem, however I don't know what to do from were you have left it, usually the ' 's would cancel out,
Sorry, could you give more clues

There's only a couple of steps left to the proof, I don't want to spoil it completely!



here's a little bit more



Now factor out and in groups.

I may have got it I think,





but can be written as as h tends to zero

thus as h tends to zero

am I correct?

(Nod)

Nice work!

wahay, kinda guessed that,
thank you very much :)