Product Rule Proof: Reasons Why it Won't Work

[UMath1]

Why wouldn't this work?

 

[Orodruin]

Because
$$
\lim_{\Delta x\to 0}\left(\frac{\Delta(g(x)h(x))}{\Delta x}\right) \neq \lim_{\Delta x\to 0}\left(\frac{\Delta g(x)}{\Delta x}\right)\lim_{\Delta x\to 0}\left(\frac{\Delta h(x)}{\Delta x}\right).
$$
It is unclear why you think these expressions would be equal on quite a few levels.

 

[UMath1]

Isn't that a limit law though?

 

[Orodruin]

You are doing something completely different from that law. Not only are you not computing ##\Delta(g(x) h(x))## correctly, you are also introducing an arbitrary extra ##\Delta x## in the denominator.

Also, you should stop attaching images like these. They are going to be impossible to read on most mobile devices and impossible to quote properly. There is a tutorial on how to write proper forum maths here: https://www.physicsforums.com/help/latexhelp/

 

[UMath1]

I understand that I put in an extra Δx. But how is Δ(g(x)h(x)) beung computed incorrectly?

 

[Orodruin]

I understand that I put in an extra Δx. But how is Δ(g(x)h(x)) beung computed incorrectly?

You are assuming ##\Delta(g(x)h(x)) = (\Delta g(x))(\Delta h(x))## to leading order in ##\Delta x##:

 

[UMath1]

Oh ok..thanks!

 

[Mark44]

Also, you should stop attaching images like these. They are going to be impossible to read on most mobile devices and impossible to quote properly. There is a tutorial on how to write proper forum maths here: https://www.physicsforums.com/help/latexhelp/

I agree completely. The image you posted in #3 can be done right here in the text input pane.

 

[Svein]

The standard proof: Let [itex] f(x)=u(x)\cdot v(x)[/itex]. Then [itex]f(x+h)=u(x+h)\cdot v(x+h) [/itex]. This means that [itex]f(x+h)-f(x)=u(x+h)\cdot v(x+h) - u(x)\cdot v(x) [/itex]. Now add and subtract [itex]u(x+h)v(x) [/itex]: [itex] f(x+h)-f(x)=u(x+h)\cdot v(x+h) - u(x+h)\cdot v(x) + u(x+h)\cdot v(x) - u(x)\cdot v(x)= u(x+h)\cdot[v(x+h)- v(x)]+v(x)\cdot [u(x+h)-u(x)][/itex].
Now divide by h and go to the limit: [itex] f'(x)= \lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}=\lim_{h\rightarrow 0}\frac{u(x+h)\cdot[v(x+h)- v(x)]}{h}+\lim_{h\rightarrow 0}\frac{v(x)\cdot [u(x+h)-u(x)]}{h}=u(x)\cdot v'(x)+u'(x)v(x)[/itex].