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I have started studying and reading proofs and tried to prove the product formula for derivatives myself. I did not know how to proceed after second step and after struggling for some time, looked at the proof in the textbook which is as follows

Product Formula

(uv)' = u'v + uv'

Proof

By definition,

$$(uv)' = \lim_{\Delta x\to 0} \frac{uv(x+\Delta x)-(uv)(x)}{\Delta x}$$ $$= \lim_{\Delta x\to 0} \frac{u(x+\Delta x).v(x+\Delta x)-u(x).v(x)}{\Delta x}$$ u(x + $\Delta x$).v(x) - u(x + $\Delta x$).v(x) = 0 and adding this to numerator won't change our equation

Thus, adding u(x + $\Delta x$).v(x) - u(x + $\Delta x$).v(x) to numerator, we get $$(uv)'=\lim_{\Delta x\to 0} \frac{u(x+\Delta x).v(x+\Delta x)-u(x).v(x)+u(x + \Delta x).v(x) - u(x + \Delta x).v(x)}{\Delta x}$$ $$=\lim_{\Delta x\to 0} \frac{(u(x+ \Delta x)-u(x))v(x)}{\Delta x}+\lim_{\Delta x\to 0} \frac{(v(x+ \Delta x)-v(x))u(x+\Delta x)}{\Delta x}$$ $$=u'(x)v(x)+v'(x)u(x)$$ $$=u'v+uv'$$

While I can see why the third step was done, how am I supposed to know what exactly to add(or multiply and divide, add and subtract, etc) to complete the proof?

I can do proofs which are straightforward but I always get stuck in proofs which require such manipulations.

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  • $\begingroup$ I'm afraid this comes down to insight. Doing and reading plenty of proofs should increase your ability. Sometimes you might not be able to prove things at first since you're missing some technique or "trick" in this case. But you might encounter similar problems to this one in the future, which you will (should) now be able to solve. $\endgroup$ – Demophilus Jun 5 '17 at 13:38
  • $\begingroup$ @Demophilus Yes. After becoming familiar with this proof, I could prove the quotient rule for derivatives (u/v)' by adding u(x)v(x) - u(x)v(x) to the numerator. But only because I am now familiar with the above proof. I am hoping for an analytical method so that I can solve unfamiliar problems without looking at the solution. $\endgroup$ – Pratik Haware Jun 5 '17 at 14:02
  • $\begingroup$ That's precisely my point. Now you know a new technique that you can use in other proofs. To be able to do more proofs, you should just try more proofs. There will always be proofs you won't be able to do from the first try. There's no "analytical method" that will give you the answer to any problem you encounter in analysis. You just have to work hard and try to learn as much as possible. $\endgroup$ – Demophilus Jun 5 '17 at 14:13
  • $\begingroup$ The big problem with most conventional proofs is that they don't give any motivating insight or intuition for the steps of the proof. The worst are "proofs without words" (not to be confused with visual proofs with diagrams) which are long sequences of equations with no reasons given .for why each step follows from previous steps. $\endgroup$ – Somos Jun 5 '17 at 21:05

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