Question

Calculate the derivative of...

Answer 1

\[\frac{ d }{ dt }\int\limits_{e^{t}}^{t ^{2}} \sqrt{3+x ^{2}} dx.\]

Answer 2

Okay.. putting that childish business aside. Do you know how to start? Any ideas?

Answer 3

I antiderive it and plug in t^(2) and subtract by e^(t) right?

Answer 4

I think there was an easier way to do this.. but I'm not sure if it applies in this particular situation

Answer 5

d/dt (F(t^2)-F(e^t))
given F'(x)=f(x)=sqrt(3+x^2)
you don't need to calculate F(t), the antiderivative
just use the chain rule

Answer 6

*F(x)

Answer 7

I doubt the fundamental theorem of calculus would be of use here?

Answer 8

d/dt (F(t^2)-F(e^t))
=F'(t^2)*2t-F'(e^t)*e^t

Answer 9

=sqrt(3+t^2)*2t-sqrt(3+e^(2t))*e^t

Answer 10

the fundamental thm of calculus, in combination with the chain rule, allows the problem to be solved without finding any anti derivatives

Answer 11

That's what I thought, thanks for the clarification @eashy

Answer 12

Hmm, I typed it in and \[2t \sqrt{3+t ^{2}}-e ^{t}\sqrt{3+e ^{2t}}\] is not the correct answer.

Answer 13

oh, i have to derive that to get the answer right?

Answer 14

i made a typo. it should be
d/dt (F(t^2)-F(e^t))
=F'(t^2)*2t-F'(e^t)*e^t
=sqrt(3+t^4)*2t-sqrt(3+e^(2t))*e^t

Answer 15

F'(t^2)=sqrt(3+t^4), not sqrt(3+t^2)

Answer 16

Ooohh, okay thank you very much!!