When you differentiate dy/dx again, you get d²y/dx², and so on with 3rd, 4th etc. Why are the exponents in different places? Meaning, why isn't it d²y/d²x or dy²/dx²?
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I'm not 100% sure on this, but I believe it comes from treating the Leibniz notation (the dy/dx notation) as a fraction. When you find the second derivative, you are finding the derivative of the first derivative. Sepecifically, you find:
d/dx dy/dx
So, just like d/dx y = dy/dx (it works kinda like a fraction), we treat the second derivative kinda like a fraction again (WARNING: The following is VERY BAD MATHS to be used for explanatory purposes only!):
d/dx * dy/dx
= (d * dy) / (dx * dx)
= d²y / dx²
d/dx dy/dx
So, just like d/dx y = dy/dx (it works kinda like a fraction), we treat the second derivative kinda like a fraction again (WARNING: The following is VERY BAD MATHS to be used for explanatory purposes only!):
d/dx * dy/dx
= (d * dy) / (dx * dx)
= d²y / dx²