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\[y^2=2px\] \[ \Rightarrow 2ydy=2pdx \] How is this achieved at? I've always gone with calculating \[ 2y\frac{dy}{dx}=2p \] and multiplying both sides by dx. But I know that's heresy. What's the official story as to how it's arrived at?

Mathematics
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I've always found it odd how it seems plucked out of thin air
here is a link this is called differentials chew on it
Oh differentials- I wasn't thinking!

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