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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?

TBBT
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    What facts are you given about the Dirac delta function? That is, what is your definition of it? Typically, those two properties are proven using "the sifting property", i.e. the fact that $\int_{-\infty }^{+\infty }f(x)\delta (x)dx=f(0)$. This is considered to be the defining property. – Ben Grossmann Apr 21 '14 at 11:41
  • @Omnomnomnom, What I mean is the book asks me to derive those properties. They want me to show that it is true. – TBBT Apr 21 '14 at 11:43
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    Can you use the fact that $\int_{-\infty }^{+\infty }f(x)\delta (x),dx=f(0)$? If not, what is the definition of the Dirac delta function? – Ben Grossmann Apr 21 '14 at 11:44
  • No, I can't. But I have just found a way to derive that one. – TBBT Apr 21 '14 at 11:47
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    What is your textbook's definition of the Dirac delta function? If we're not starting from the sifting property, where are we starting? – Ben Grossmann Apr 21 '14 at 11:50
  • I think I got $\int_{-\infty }^{+\infty }f(x)\delta (x)dx=f(0)$ covered. Thank you for telling me about sifting property. I only have $\int_{-\infty }^{+\infty }f(x)\delta '(x-y)dx=-f'(y)$ and $\int_{-\infty }^{+\infty }f(x)\delta (x-y)dx=f(y)$ to finish. – TBBT Apr 21 '14 at 11:54
  • My book definition of DDF is the $\int_{-\infty }^{+\infty }f(x)\delta (x)dx=f(0)$. I think they kind of like OK, I introduced you about definition, and now you learn how to derive it. – TBBT Apr 21 '14 at 11:57
  • Oh, okay, that's what I expected. To show that $\int_{-\infty }^{+\infty }f(x)\delta (x-y),dx=f(y)$, it specifically helps to make the $u$-substitution (change of variables) $u = x-y \implies du = dx$ – Ben Grossmann Apr 21 '14 at 11:59

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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.

Ben Grossmann
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