Derivative of the composition of delta distribution with a differentiable function

Question

Is there an explicit representation of what $\frac{\partial }{ \partial x} \delta(f(x,t))$ is?

Here $\delta $ is the delta distribution and $f$ is an arbitrary differentiable function which is composed with $\delta$.

Answer 1

The derivative for distribution can only be seen as in the test function. That is, $$\langle\frac{\partial }{\partial x}\delta(f(x,t)),\phi(x,t) \rangle = -\langle \delta(f(x,t)),\frac{\partial }{\partial x}\phi(x,t)\rangle=-\sum_{\{(x,t)|f(x,t)=0\}}\frac{\partial }{\partial x}\phi(x,t)$$