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# C4 Parametric Differentiation Watch

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1. I learned in C3 that one version of the chain rule is . There was no proof for it, which is annoying, so if anyone can prove it with A level knowledge that would be helpful.

Now in C4, I've been told that I can use the chain rule, but rearranged into the form . Can somone tell me how they arrived at this result. I know dividing is the same as multiplying by the reciprocal, but surely it can't be the same for derivatives? And if it is, then why didn't they stick to the result taught in C3?
2. (Original post by ViralRiver)
I learned in C3 that one version of the chain rule is . There was no proof for it, which is annoying, so if anyone can prove it with A level knowledge that would be helpful.

Now in C4, I've been told that I can use the chain rule, but rearranged into the form . Can somone tell me how they arrived at this result. I know dividing is the same as multiplying by the reciprocal, but surely it can't be the same for derivatives? And if it is, then why didn't they stick to the result taught in C3?
The proof of

is no different to the proof of

Which I suppose answers both of your questions, it's not special just because it's a derivative.
3. Hmm okay, my teacher said that you could think of it as the 'du's cancelling out, but that's not a proper mathematical explanation, which is what led me to this topic,
4. (Original post by ViralRiver)
Hmm okay, my teacher said that you could think of it as the 'du's cancelling out, but that's not a proper mathematical explanation, which is what led me to this topic,
If you've got a parabola, obviously

Parametrically,

Using the , you can quite easily derive

Using the parametric equations you get

Which, using

produces

Then using the prior relationship between x and t, you get

Since

So you are essentially cancelling out the variable t.
5. Actually scrap that above, I just worked out a much nicer 'proof'.

Parametrically.

Going back to the original equations you can see that since

and

and since

(chain rule)

Plugging that all back into

gives

Cancelling the terms gives

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Updated: November 28, 2010
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