Thread: Leibniz formula and chain rule

Hi,

I've been learning Maths by myself and I got stuck with the Leibniz formula. The theory is as follows:
"Lets suppose that both partial derivatives of f(t,x) exist and that a(t) and b(t) are derivable functions and F (t).

F (t) = ∫b(t)a(t) f (t, x) dx for all t.

Then F '(t) = f (t, b(t))b'(t) − f (t, a(t))a'(t) + ∫b(t)a(t) f 1'(t, x) dx.

So, I don´t completely understand the formula and I got my exercises wrong once and again. Please, could you explain step by step how to do it? I'll write you below and exercise and my attempt to get a) done.

For example: Find F'(t) in the following examples:

a) F (t) = ∫2tt x2 [squared x] dx .
b) F(t)= ∫21 (etx)/x dx

What I did in a)

F'(t)= (2t)2*2 - (t)2 + ∫2tt 2x dx -->
--> 4t2*2 - t2 + ∫2tt 2x dx
--> 7t2 + (2t)2 - (t)2
--> 4t2
ANSWER IN THE BOOK:
F'(t)= (2t)2*2 - (t)2 = 7t2

ANY HELP WOULD BE REALLY APPRECIATED.

THANK YOU

Please look here
Differentiation under the integral sign - Wikipedia, the free encyclopedia
Could you look Latex help in the main window of MHF.
I can't understand what is written in (a).