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I have a serious difficulty to prove the following identities. Thank you in advance to help me please.

Let $\varphi$ be a test function on $\mathbb{R}$.

Let $x, y \in \mathbb{R}^n$. We denote $\tau_x \tilde{\varphi}(y)= \varphi(x-y)$. ($\tilde{\varphi}(x)= \varphi(-x))$.

My question is: let $\alpha \in \mathbb{N}^n$. How can we prove that $$ D_y^\alpha(\tau_x \tilde{\varphi})= (-1)^{|\alpha|} \tau_x(D_y^{\alpha} \tilde{\varphi})= (-1)^{|\alpha|} D_y^{\alpha}(\tau_x \tilde{\varphi}) $$ ?

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    What is $\varphi$? Test function or distribution? Did you try to show these identities by definition? Where do you get stuck? – 2017-02-10
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    sorry$ $\phi$ is an function test, i edit my post. I try but i haven't any idea, please to help me – 2017-02-10
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    If $\varphi$ a test function (in other words, a $C^\infty$ function with compact support), then everything can be done by the definition of a derivative. Start by studying $\nabla_x \varphi(x-y)$ and $\nabla_y \varphi(x-y)$ – 2017-02-10

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