Indeed, the equality will not hold in general. For counterexamples, see this or this.

For sufficient conditions for the equality when $d=1$, see e.g. Folland, Theorem 2.27 or the more general Lemma 2.3. This immediately extends to any $d$ if the derivatives are understood in the Gateaux sense. As seen from the discussion of Theorem 2.27 in Folland, the extension to $d>1$ is somewhat more problematic if the derivatives are understood in the Fréchet sense.

Indeed, the equality will not hold in general. For counterexamples, see this or this.

For sufficient conditions for the equality when $d=1$, see e.g. Folland, Theorem 2.27 or the more general Lemma 2.3. This immediately extends to any $d$ if the derivatives are understood in the Gateaux sense. As seen from the discussion of Theorem 2.27 in Folland, the extension to $d>1$ is somewhat more problematic if the derivatives are understood in the Fréchet sense. This answer may also be of interest to you.

Indeed, the equality will not hold in general. For counterexamples, see this or this.

Indeed, the equality will not hold in general. For counterexamples, see this or this.

For sufficient conditions for the equality when $d=1$, see e.g. Folland, Theorem 2.27 or the more general Lemma 2.3. This immediately extends to any $d$ if the derivatives are understood in the Gateaux sense. As seen from the discussion of Theorem 2.27 in Folland, the extension to $d>1$ is somewhat more problematic if the derivatives are understood in the Fréchet sense.