# Thread: simple derivative but confusing me

1. ## simple derivative but confusing me

without evaluating the integral, is the derivative of the function $\displaystyle \int_{2}^{x} e^{t^{2}-2} dt$ just $\displaystyle e^{t^{2}-2}$?

Thanks in advance for any assistance recieved

2. ## Response

Originally Posted by tsal15
without evaluating the integral, is the derivative of the function $\displaystyle \int_{2}^{x} e^{t^{2}-2} dt$ just $\displaystyle e^{t^{2}-2}$?

Thanks in advance for any assistance recieved
You have the general form correct, but you would have to evaluate the function as follows:

e^(x^2 - 2)

3. Originally Posted by ajj86
You have the general form correct, but you would have to evaluate the function at its endpoints.

So, it would be F(x) - F(2), or

e^(x^2 - 2) - e^(2^2 - 2)
e^(x^2 - 2) - e^2
Hey ajj86,

First I gotta say, you've been very helpful within the period of 20mins, more than some others on this forum have been in 20days.

MMM, I believe that you are implying the fundamental theorm of calculus whilst using antidifferentiation techniques aka finding the area under graph aka evaluating the integral...? I'm aware it would be a more perfect answer (can u get more perfect than perfect?), but the question strictly states that i must: "Without evaluating the integral, find F'(x) where F(x) = $\displaystyle \int_{2}^{x} e^{t^2-2} dt$" so having said that would $\displaystyle e^{t^2-2}$ be the answer?

Thank you

tsal15

4. ## Response

I just found this:

But, I'm not really sure as to whether the answer I gave is correct. Let me do a little more investigating.

5. Originally Posted by ajj86
I just found this:

But, I'm not really sure as to whether the answer I gave is correct. Let me do a little more investigating.

ok. no problems. I'll wait anxiously

6. ## Response

According to the form given, I'm thinking the answer would be:

e^(x^2 - 2)

7. Originally Posted by ajj86
According to the form given, I'm thinking the answer would be:

e^(x^2 - 2)

ok. 2 things

1. Did u evaluate the integral or how did u do this?

2. why is it not $\displaystyle e^{t^2-2}$

8. ## Response

Tsal15, I apologize for not replying last night. I'm not trying to take the easy way out, but I think this might give a better explanation than I can give:

Calculus Facts: Derivative of an Integral

Let me know what you think.

9. This is just a simple application of the Second Fundamental Theorem of Calculus

$\displaystyle \frac{d}{dx}\int_a^{g(x)}f(t)dt=f(g(x))\cdot{g'(x) }$

The proof of this is most likely beyond the scope of your class.

10. Originally Posted by Mathstud28
This is just a simple application of the Second Fundamental Theorem of Calculus

$\displaystyle \frac{d}{dx}\int_a^{g(x)}f(t)dt=f(g(x))\cdot{g'(x) }$

The proof of this is most likely beyond the scope of your class.
I'm sure of it but could you show me anyways... I'd like to learn something outside the class rather than just from the tutor... thanks

11. Originally Posted by ajj86
Tsal15, I apologize for not replying last night. I'm not trying to take the easy way out, but I think this might give a better explanation than I can give:

Calculus Facts: Derivative of an Integral

Let me know what you think.

Thanks that was quite helpful. I'll be sure to recommend you to my girlfriends...i think they need more help than i do hehehehe

well, thanks again, ajj86

12. Originally Posted by tsal15
I'm sure of it but could you show me anyways... I'd like to learn something outside the class rather than just from the tutor... thanks
Wikipedia actually has a pretty non-analysis friendly version. Look here

Fundamental theorem of calculus - Wikipedia, the free encyclopedia