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Is (dx²)/(d²y) is acceptable as second derivative?

I have just read about derivative that on higher derivative is [d^(n)y/dx^(n)]=y'(n) and also the inverse function (1/y')=x'

But i am still confuse if it is possible
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John Sebastian
Asked:
John Sebastian
1 Solution
 
d-glitchCommented:
What are your reading and why?

From Wikipedia:   https://en.wikipedia.org/wiki/Derivative
The dy/dx notation is from Leibniz.  Note that the y term is usually on top.
The ' (or prime) notation is from Lagrange.  Both notations are in common use.

I don't understand you question about an "inverse function."
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Ray PaseurCommented:
How is this a PHP question? Are you looking for a code sample in PHP?  Or C++?  Please clarify, thanks.
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d-glitchCommented:
>> JS
     Comments and follow-on questions should be posted here not in personal messages.

We can not see the context for the disagreement between you and your colleague.  
What is the specific question that caused the disagreement, and what are the competing answers?

I have already noted that your expression is flipped from the usual form of the 2nd derivative:
    (d²y)/(dx²)    <== Why do you think that this form is inadeqauate?
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John SebastianAuthor Commented:
It is because i want to prove it using the inverse function that dy/dx=1/(dx/dy) and dx/dy=1/(dy/dx)
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d-glitchCommented:
I still don't see a context.  You have mentioned 2nd derivatives, 1st derivatives, inverse functions, and the reciprocal function (1/x).

I don't see a clear starting point, or a path to any useful result.
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BigRatCommented:
Conventionally the dash on a variable means differentiation with respect to time, better said differentiation with respect to t.

Thus y' is dy/dt and x' is dx/dt.

The question is whether d²x/dy² an acceptable second differential and the answer is YES if, and strictly speaking only if, the function is x=f(y). This is because the "²" is written between the "d" and the "x" on the top and the "²" is written after the "y" on the bottom.

Now we could have a function y=F(x) and form the second differential d²y/dx² by differentiating F twice, and, under certain circumstances, the result of differentiating F twice will be the same as the result of differentiating f twice and inverting the result. But a function like y=x has no second differential, for dy/dx is 1 and d²y/dx² is then zero, so d²y/dx² would be 1/0 which is not possible. So a function like x=y+2 has no second differential and so it's inverse won't exist.

There is also a class of non-differentiable functions discovered by Weierstrass, so one has to be a bit careful on taking differentials.
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d-glitchCommented:
The questions remains unclear, but BigRat's answer is comprehensive.
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