# Thread: interchanging the order of integral and derivative

1. ## interchanging the order of integral and derivative

could someone explain why the following equation is incorrect?

$\displaystyle \frac{d^2}{dx^2}\int^1 _{-1} \log|x-t|dt$=$\displaystyle \int^1_{-1}\frac{d^2}{dx^2}\log|x-t|dt$=$\displaystyle \int_{-1}^1 \frac{-1}{(x-t)^2}dt$

2. Originally Posted by Kat-M
could someone explain why the following equation is incorrect?

$\displaystyle \frac{d^2}{dx^2}\int^1 _{-1} \log|x-t|dt$=$\displaystyle \int^1_{-1}\frac{d^2}{dx^2}\log|x-t|dt$=$\displaystyle \int_{-1}^1 \frac{-1}{(x-t)^2}dt$
Is $\displaystyle \frac{\partial}{\partial t}\ln|x-t|$ even defined for $\displaystyle t\in[-1,1]$ and $\displaystyle x\in[-1,1]$?

3. Originally Posted by Drexel28
Is $\displaystyle \frac{\partial}{\partial t}\ln|x-t|$ even defined for $\displaystyle t\in[-1,1]$ and $\displaystyle x\in[-1,1]$?
sorry i dont understand. the derivatives in the equation are with respect to x but not t.

4. Originally Posted by Kat-M
sorry i dont understand. the derivatives in the equation are with respect to x but not t.
Look here