# Distribution - Heavisidefunction - Second Derivative

• March 15th 2011, 09:12 AM
4Math
Distribution - Heavisidefunction - Second Derivative
Hi,

I have the following function:

$\display\ f(t)= t \theta (t) + (\frac{1}{2}t^{2}-t+\frac{1}{2})\theta (t-1)+(t-2)\theta (t-2)+(-\frac{1}{4}t^{2}+\frac{5}{2}t-\frac{21}{4})\theta (t-3)+(-\frac{1}{4}t^{2}+3t-8)\theta (t-4)$

I need the second derivate $f''(t)$

How do I do this? Do I use the product rule like if would have been a "regular" function??

$\display f'(t)=1\cdot\theta (t)+t\delta(t)$
$\display+(t-1)\theta (t-1)+(\frac{1}{2}t^{2}-t+\frac{1}{2})\delta(t-1)+$
$\display+\theta(t-2)+(t-2)\delta(t-2)+$
$\display+(-\frac{1}{2}t+\frac{5}{2})\theta(t-3)+(-\frac{1}{4}t^{2}+\frac{5}{2}-\frac{21}{4})\delta(t-3)+$
$+(-\frac{1}{2}+3)\theta(t-4)+(-\frac{1}{4}t^{2}+3t-8)\delta(t-4)$

Is $f'(t)$ correct?

Any guidance would be greatly appritiate it.
Thank you