# Differentiating delta function composed with a function

1. Jun 21, 2013

### rms502

Dear all,
I just wondered whether there was any standard identity to help me solve this equation:
$$\int \delta(f(x))^{\prime\prime}g(x) dx$$

2. Jun 21, 2013

### mathman

You don't have an equation.

I am not sure what you mean by the derivative of the delta function (derivative with respect to what?).

Integration by parts twice might be an approach.

3. Jun 21, 2013

### lurflurf

Several things to consider
$$\delta(\mathop{f}(x))''=\mathop{f}''(x) \delta (x)+(\mathop{f}'(x))^2 \delta '' (x) \\ \int \! \delta ^{(n)} (x) \, \mathop{f} (x) \, \mathop{dx}=(-1)^n\int \! \delta (x) \, \mathop{f ^{(n)}} (x) \, \mathop{dx}\\ \int \! \delta (\mathop{f} (x)) \, \, \mathop{g} (x) \mathop{dx}=\sum_{x \in f^{-1}(0)} \mathop{g}(x)$$