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As everyone knows that the derivitive of a function is notated as $\frac{dy}{dx}$

The question is: How is the second derivitive $\left(\frac{d^2y}{dx^2}\right)$ notation derived?

Itakura
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  • That's a notation for the second derivative. –  Feb 14 '16 at 01:20
  • http://math.stackexchange.com/questions/852394/failure-of-differential-notation –  Feb 14 '16 at 01:37
  • Is the question asking about the definition of a second derivative, the rationale for the notation for a second derivative, or for some historical perspective on such a notation? As above, the Question is unclear in its present statement. – hardmath Feb 14 '16 at 03:56
  • Think of the derivative of being two operations rather than one---a differential followed by a division by dx. Now the differential of a variable is d(varname) or just dvarname. The differential applied twice is d(d(varname)), or, as a convenience, this can be written as d^2 varname.

    Therefore, since the derivative is a differential followed by a division by dx, it is d^2y/dx^2 because you are doing the differential of y twice, and dividing by dx twice.

     – johnnyb Feb 14 '16 at 07:23

2 Answers2

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This is just notation. Let $y=f(x)$. Then $\frac d{dx}(y)=f'(x)$. Now $$f''(x)=\frac d{dx}\left(\frac d{dx}(y)\right)=\left(\frac d{dx}\right)^2(y)=\frac {d^2}{dx^2}(y).$$

Tim Raczkowski
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  • so $\frac{d}{dx}\cdot \frac{dy}{dx} = \frac{d^2y}{dx^2}$. Ok its not a fraction, guess I have to accept this explanation. – Itakura Feb 14 '16 at 01:33
  • @KennyGuy The notation is confusing: These manipulations are not really rigorous, else you could rearrange it to get things like $d\cdot \frac {dy}{dx^2}$ which makes no sense. – YoTengoUnLCD Feb 14 '16 at 01:43
  • The idea is that $d/dx$ is an operation on functions which sends a function to it's derivative. $(d/dx)^2$ would denote applying the operator twice in succession. – Tim Raczkowski Feb 14 '16 at 01:48
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    @YoTengoUnLCD You are absolutely right about rigor here. This is not so much a derivation as an explanation of the notation. – Tim Raczkowski Feb 14 '16 at 01:50
  • Yes, I just wanted OP not to feel deceived by the notation, in any way, we could just say $\frac {d^2}{dx^2}$ looks much better than $\frac d {dx} \frac d{dx}$ :-P (although the operator approach is much believable). – YoTengoUnLCD Feb 14 '16 at 01:53
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Edit: I misread the OP question and thought that practical examples were asked for.

The second derivative arises from considering the first derivative as an independent function, and is obtained by derivation. Notation (like the other answerer mentioned), is mainly of two types: Leibniz notation or: $ {d\over dx} (f(x))$ is the first order derivative ${d\over dx} ({d\over dx}(f(x)) = {d^2\over dx}f(x)$

Lagrange Notation instead uses "prime" marks: Derivative of $f(x) = f'(x)$, derivative of $f'(x) = f''(x)$ and so on.

Practical example: $$ f(x) = x^3 $$ $$ f'(x) = {d\over dx}f(x) = 3x^2 $$ $$ f''(x) = {d\over dx}f'(x) = 6x $$ A pratical example is however the change in kinetic energy of a moving body.

Morgormir
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  • That's not really the question (if I understood correctly, the OP is asking about the notation, not the meaning or use of the second derivative). – Clement C. Feb 14 '16 at 01:25
  • Fair, but a use of the second derivative in a practical application is in fact the example I gave. Edit: Misread original post, will change answer to reflect it. – Morgormir Feb 14 '16 at 01:26