dirac delta – Derivative of the probability transformation formula

Let $ f: mathbb {R} ^ n rightarrow mathbb {R} $ to be a measurable function of Borel. (It can further be assumed that $ | frac { partial f} { partial z_i} |> $ 0.

$ overrightarrow {Z} = (Z_1, Z_2, points, Z_n $ is a random variable vector with probability density function $ p _ { overrightarrow {Z}} $. Let $ Y: = f ( overrightarrow {Z}) $ be a random variable with probability density function $ p_ {Y} $.

By the probability transformation formula, we have

$ p_Y (y) = int _ { mathbb {R} ^ n} p _ { overrightarrow {Z}} ( overrightarrow {z}) delta (yf ( overrightarrow {z}) d overrightarrow {z } $.

We want to know $ frac {dp_Y} {dy} $? I do not really know how we take the derivative to the $ delta (y-f ( overrightarrow {z}) $ as $ y $ also depends on $ overrightarrow {z} $.

Moreover, we can assume that $ Z_i $ is i.i.d, so we could also have
$$ p_Y (y) = int _ { mathbb {R}} p_z (z_n) dz_n dots int _ { mathbb {R}} p_z (z_1) delta (yf ( overrightarrow {z}) dz_1 $$.

Someone has any idea?