# Help with a derivative

1. Sep 14, 2010

### xchangz

2. Sep 14, 2010

### Staff: Mentor

What have you tried? Before we can give you any help, you must show some effort at solving your problem.

3. Sep 14, 2010

### xchangz

Well what I've tried so far is that I know that the fundamental theorem of calculus derivative with respect to t of integral from 0 to t f(x) dx is just f(t). However, that won't work in this case because we have the introduction of a 2nd variable t. It won't be just f(t,t). If I assume f(t,x) = t^2 +3tx + x^2 just as a dummy definition as an experiment, I get that g'(t) = 17t^2 / 2 but if we assume it is f(t,t), it will be 5t^2 so the two answers don't match up. I've been staring at this problem for hours now!

4. Sep 14, 2010

### gabbagabbahey

Look under "variable limits" here. You basically want to combine the usual FTC with the chain rule (since there is $t$-dependence in both the limits and the integrand).

5. Sep 14, 2010

### xchangz

Thanks a lot! But I do not totally understand the proof. Under the general form of variable limits, which is similar to my problem, I understand everything up until the part with the mean value theorem. I don't understand what went on there. Though this provides me with a better insight on the problem.