# Differential of this

Define a operation

$(\partial \cdot x)*f(x)=\frac{d(xf(x))}{dx}$

In the notes it has that

$(\partial^n \cdot x)*f(x)=(n \partial^{n-1 }+ x \cdot \partial^n)*f(x)$

Is this wrong? I can't get this

Well, $(\partial^n \cdot x)*f(x)=\frac{d^{n-1}}{dx^{n-1}}(\frac{d(xf(x))}{dx})$

Which, doesn't seem to get that.

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What is the symbol you're defining a meaning for -- is it "$\partial\cdot$", or "$*$", or what? –  Henning Makholm Nov 27 '11 at 0:13

The derivative of $xf$ is $f+xf'$. The 2nd derivative is $2f'+xf''$. The 3rd derivative is $3f''+xf'''$. This seems to me to be in accord with the formula in the notes.