Proof of an identity of the Bessel function

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Could someone just give me a hint for this one? I'm simply stuck on how to get anything out of the grips of that derivative:

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For reference:

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Also, why can't we simplify this part:

$$x^n \left ( -\frac{1}{x}\frac{d}{dx} \right )^n \to \left ( -\frac{d}{dx} \right )^n$$

??