Differential NotationDate: 04/16/2009 at 05:50:59 From: Jordan Subject: Differential Notation Why is the second derivative written as d^2y/dx^2? Date: 04/16/2009 at 13:54:14 From: Doctor Vogler Subject: Re: Differential Notation Hi Jordan, Thanks for writing to Dr. Math. It seems mysterious how the numerator repeats the d but the denominator repeats the dx, but it makes sense if you think about it the right way. You are familiar with the notation dy -- dx to mean "the derivative of y with respect to x". Sometimes, when you want to use a formula instead of a variable, you can put that in the place of y, as in d(cos x) -------- dx or, more conveniently (especially for complicated formulas), d --(cos x). dx So then if that is the derivative of cos x, then what is its derivative? Naturally, it would be d d --( --(cos x) ). dx dx Does that make sense? And when you write it this way, it is natural that one would abbreviate this by using the "squaring" notation on the d in the numerator and the dx in the denominator, as in d^2 ----(cos x). dx^2 Of course, you're not really squaring, because it's not multiplication; but it's a convenient notation, especially when the 2 becomes 12 or an unspecified n. Similarly, the derivative of dy/dx would be d dy --(--) dx dx which is why you get d^2y/dx^2. I should like to point out that we are *NOT* squaring the x in the denominator, like d/d(x^2), but are squaring the dx. It might have been better if we wrote (dx)^2 instead of dx^2, but that is not the notation in common usage by mathematicians, who instead treat the "dx" as a single piece. Does this explanation make sense? I hope I have cleared things up for you. If you have any questions about this or need more help, please write back, and I will try to offer further suggestions. - Doctor Vogler, The Math Forum http://mathforum.org/dr.math/ Date: 04/16/2009 at 16:32:20 From: Jordan Subject: Thank you (Differential Notation) Comprehensive and laconic answer - thanks ever so much! |
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