If we have the function(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

f : x \mapsto f(x) = 3x^2,

[/tex]

I am used to Lagrange's prime notation for the derivative:

[tex]

f' : x \mapsto f'(x) = 6x.

[/tex]

I'm fond of this notation. But it has been mostly abandoned in my engineering courses in favor of Leibniz's notation, using differential operators such as [itex]\frac{\mathrm{d}}{\mathrm{d}x}[/itex] per the above. I want to express the same relation as the second equation above using Leibniz's notation, but am having a lot of trouble. In particular, I understand that the derivative operator is a function that maps from a set of functions to a set of functions, so if

[tex]

\frac{\mathrm{d}}{\mathrm{d}x} : f \mapsto \frac{\mathrm{d}}{\mathrm{d}x}f

[/tex]

is true, then it seems correct (to me) to write

[tex]

\frac{\mathrm{d}}{\mathrm{d}x}f : a \mapsto \frac{\mathrm{d}}{\mathrm{d}x}f(a),

[/tex]

but I've been told that in order to show that a differentiated function is evaluated at some arbitrary point such as [itex] a [/itex], it is necessary to write something like

[tex]

\frac{\mathrm{d}}{\mathrm{d}x}f(x)\vert_{x=a},

[/tex]

which means that the mappings as I have them are not correct, and it would be more correct to write something like

[tex]

\frac{\mathrm{d}}{\mathrm{d}x} : f(x) \mapsto \frac{\mathrm{d}}{\mathrm{d}x}f(x)

[/tex]

and

[tex]

\frac{\mathrm{d}}{\mathrm{d}x}f(x) : a \mapsto \frac{\mathrm{d}}{\mathrm{d}x}f(x)\vert_{x=a}.

[/tex]

Any help would be greatly appreciated; I'm starting to confuse myself. I am just trying to express the derivative of a function using Leibniz's notation in arrow notation.

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# Function notation for the derivative operator

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