What is a derivative in the distribution sense?

[pellman]

Never mind. I got this one. Couldn't figure out how to delete the post though.

 

[micromass]

Let $u\in L^2(\mathbb{R})$ be a function such that for all smooth $\psi\in \mathcal{C}_c^1(\mathbb{R})$, we have that
\int_{-\infty}^{+\infty} \psi(x) u(x)dx = -\int_{-\infty}^{+\infty} \psi'(x) \varphi(x)dx
Then $u$ is said to be the derivative of $\varphi$ in the distributional sense.

 

[pellman]

Let $u\in L^2(\mathbb{R})$ be a function such that for all smooth $\psi\in \mathcal{C}_c^1(\mathbb{R})$, we have that
\int_{-\infty}^{+\infty} \psi(x) u(x)dx = -\int_{-\infty}^{+\infty} \psi'(x) \varphi(x)dx
Then $u$ is said to be the derivative of $\varphi$ in the distributional sense.

Thanks, micromass! I had just found the answer and was editing the OP. Still much appreciated though.