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I have a question I've been wondering about for a long time.

Is operation $«\color{red}{\frac{d}{dx}}»$ mathematically equal to operation $«\color{red}{'}»$ ?

Is there any difference between them?

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    No difference, just different notations for the same thing. Something similar was already asked https://math.stackexchange.com/questions/1966777/newton-vs-leibniz-notation – Virtuoz Aug 19 '19 at 14:07
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    For functions not using $x$ as its variable, one needs to either change $x$ in $\frac{d}{dx}$ or use $'$ when talking about its derivative. See also https://en.wikipedia.org/wiki/Notation_for_differentiation –  Aug 19 '19 at 14:12
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    One is a noun (') and the other a verb ($\frac{d}{dx}$). – Sean Roberson Aug 19 '19 at 14:25

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There is no difference between the notations - they mean exactly the same thing. However, at different times you will find one more useful than the other. For example, when doing u-substitution with integrals, the $\frac{d}{dx}$ is helpful. The same thing is true when using the chain rule - it is often easier to keep track of what is happening with $\frac{d}{dx}$. But, writing " ' " is definitely quicker, more efficient, and sometimes neater.

Burt
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There is no mathematical difference. They are different notations for the same thing. (The first was due to Newton, the second to Leibniz.)

user247327
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    The prime notation is due to Lagrange. Newton didn’t use the function notation introduced by Euler. – egreg Aug 19 '19 at 14:30
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Using prime ' usually indicates that we take the total derivative w.r.t. to all variables, whereas $\frac{d}{dx}$ indicates that we take the total derivative with respect to the variable $x$. Note that this is different from the partial derivative $\frac{\partial }{\partial x}$.

As an example, consider $f(x, y, z)=x^2+y^2+3z$ where $y=\sin(x)$. Then

  • $f' = (2x, 2y, 3) $ (the Jacobian matrix whose entries are the partial derivatives)
  • $\frac{d}{dx}f = 2x + 2y\cos(x) $
  • $\frac{\partial}{\partial x}f = 2x$
Hyperplane
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