Direct Delta Function

Question

I was reading a Mathematics for Physics book when I saw these exercises. By using the knowledge of direct delta function, show that:

$\int_{-\infty }^{+\infty }f(x)\delta '(x-y)dx=-f'(y)$

$\int_{-\infty }^{+\infty }f(x)\delta (x-y)dx=f(y)$

I have been working on those for quite sometime, but I can't solve simply because I don't understand much about direct delta function. Could you help me please?

Answer 1

Hint: Using the fact that $\int_{-\infty }^{+\infty }f(x)\delta (x)\,dx=f(0)$, show that the second equality holds by using $u$-substitution. Then, show that the first equality holds using integration by parts.