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Very simple question, I didn't understand the way we are using derivative operator(dy/dx) when we want to derive multiple times such as $d^{10}f(x)/dx^{10}$ for a function such as;

$$f(x)=x^{12}-4x^{3}-4$$

Okey I get it when I want to derive a function 10 times I use $d^{10}f(x)$ but why I am using $dx^{10}$ at the bottom of the operator instead of $d^{10}x$ because I am not deriving for ''$x^{10}$'' variable but for just ''$x$'' variable for the each ''$x$'' in my function not for each ''$x^{10}$'' I would love to hear an explanation about usage of this because it was so quick and basic in my lesson and I feel like this is important where I put these exponential expressions in the derivative operators.

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This is just a matter of notation. You should read $$\left(\frac{d}{dx}\right)^{10}=\frac{d^{10}}{(dx)^{10}}$$ and this is usually written as $\frac{d^{10}}{dx^{10}}$, avoiding the parenthesis in the denominator.

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  • $\begingroup$ Oh now I get it thank you very much sir. $\endgroup$ – E.Berk Feb 26 at 9:06

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