Re: Proving the Product Rule

first of all, i am confused by the definition of . as i see it, this should be:

that is, we want to represent the change in h between the points and (x_{0} plus "a little bit more (plus/minus)"). what you have doesn't look right at ALL, since as , but the idea should be: "h changes very little, if x does".

writing similar definitions for :

therefore:

this is almost what we want, the only "sticking point" is the term , so to finish the proof, one has to show that:

so that one can conclude:

Re: Proving the Product Rule

I'm sorry! That was a really silly mistake on my part. Embarrassing stuff. I really appreciate you taking the time to correct me.

Re: Proving the Product Rule

Quote:

Originally Posted by

**Ivanator27** I'm sorry! That was a really silly mistake on my part. Embarrassing stuff. I really appreciate you taking the time to correct me.

You can also write it as

Because must be continuous we have .

That gives you the result at once.

Re: Proving the Product Rule

Quote:

Originally Posted by

**Plato** You can also write is as

Because

must be continuous we have

.

That gives you the result at once.

the continuity of g is also why the term vanishes, since:

and recalling that:

and *because g is continuous*

(the continuity of g is essential, if g suddenly "jumps" at x_{0}, we cannot say that g(x_{0}+Δx) is anywhere near g(x_{0})).

EDIT: the Wikipedia page : Product rule - Wikipedia, the free encyclopedia has several different proofs, and some nice pictures, too.

Re: Proving the Product Rule

Quote:

Originally Posted by

**Deveno** the continuity of g is also why the term

vanishes, since:

and recalling that:

and *because g is continuous*

(the continuity of g is essential, if g suddenly "jumps" at x

_{0}, we cannot say that g(x

_{0}+Δx) is anywhere near g(x

_{0})).

My problem with that is simple: I have no idea how is defined.

Whereas clearly is defined as a variable, what is

Re: Proving the Product Rule

Quote:

Originally Posted by

**Plato** My problem with that is simple: I have no idea how

is defined.

Whereas clearly

is defined as a variable, what is

as indicated in post 2, for any function f:

this is a function of the variable Δx (in it, x_{0} plays the role of a constant).