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How to write the expression for first order differential operator in Mathematica? What I mean to say that I want to evaluate

$$\frac{\partial}{\partial x}f(x)=f'(x)+f(x)\frac{\partial}{\partial x}.$$

I am struggling with the last term format i.e., $\frac{\partial}{\partial x}$.

Example:

$$ \frac{\partial}{\partial x}e^{ax}=ae^{ax}+e^{ax}\frac{\partial}{\partial x}$$

where $a=a(y)$ and $\frac{\partial}{\partial x}$ is first order differential operator.

Please find the definition of first order differential operator here in the accepted answer: Do derivatives of operators act on the operator itself or are they "added to the tail" of operators?

Sam
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  • An operator needs something to operate on, right? So what does $\frac{\partial}{\partial x}$ on its own mean, computationally wise? – Nasser Mar 16 '22 at 07:18
  • https://physics.stackexchange.com/questions/55773/do-derivatives-of-operators-act-on-the-operator-itself-or-are-they-added-to-the Got the definition from the given link. Please check the accepted answer. – Sam Mar 16 '22 at 07:21
  • Check the notebook or pdf here, subsection "Some noncommutative algebraic manipulation". The first example defines a differentialOperate function that I think has the features you want. – Daniel Lichtblau Mar 16 '22 at 13:21
  • @DanielLichtblau Will you please comment on this link https://mathematica.stackexchange.com/questions/265866/second-order-differential-operator-in-mathematica – Sam Mar 29 '22 at 07:33

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