Compute the derivative of the Dirac expectation of a norm

Question

I want to compute the derivative of $R$ with respect to $a$

$$ R = \int\delta_0(x) || \nabla_x(ax+b)||^2 dx$$

I am not sure what to do. Whenever I start I get a weird result. For example: $$ \nabla_x(ax+b) = a $$

$$ ||\nabla_x(ax+b)||^2 = a $$ $$ R=a $$ Can someone please correct me?

Thanks

Answer 1

I would see it like that: \begin{align} R &= \int\delta_0(x) ||\nabla_x(ax+b)||^2\,\text{d}x=||a||^2\int\delta_0(x) \,\text{d}x=||a||^2\\ \frac{\partial R}{\partial a}&=2||a||\frac{\partial ||a||}{\partial a}=2||a||\frac{a}{||a||}=2a \end{align}