Meaning of Second Derivative Notation
Date: 07/08/2004 at 16:44:45 From: Jamie Subject: second derivative notation What does the second derivative notation, (d^2*y)/(d*x^2) really mean? I understand that the notation in the numerator means the 2nd derivative of y, but I fail to understand the notation in the denominator. Isn't it supposed to mean with respect to x? Why is there an x^2 in the notation?
Date: 07/08/2004 at 23:39:22 From: Doctor Peterson Subject: Re: second derivative notation Hi, Jamie. I don't think this is explained nearly as often as it should be! There is no x^2 in this notation, and in fact no multiplication (ie, it is _not_ d*x^2 as you say). It is d^2y ---- dx^2 and the "d" represents the "differential operator", which evidently has higher precedence than exponentiation. That is, "dx" as a whole is thought of as a quantity (think of it as a small change in x), and the denominator is "(dx)^2". But here is where it comes from: the second derivative is just the derivative of the derivative, or d dy d(dy) d^2y --(--) = ------ = ---- dx dx (dx)^2 dx^2 You might read it as "the second derivative of y, with respect to x TWICE"; that last word is the reason for the "dx^2". When you have functions of more than one variable you can see things like d^2z ----- dx dy (though a modified "d" is used to avoid some confusion); this means you are taking one derivative with respect to x and another with respect to y: d dz --(--) dx dy This notation is based on analogies to fractions, and it can be dangerous to imagine that the dx and dy and d alone actually stand for numbers; but the notation works very well in making many formulas memorable. See this page for more on differentials: Differentials http://mathforum.org/library/drmath/view/53678.html If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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