MHB Solve Derivative of Integral with F(x) - SnowPatrol Yahoo Answers

AI Thread Summary
The discussion revolves around finding the derivative of the integral function F(x) defined as the integral of e^(t^2) from sin(x) to cos(x). The user initially struggles with the integration and differentiation process, leading to confusion about the application of the Fundamental Theorem of Calculus (FTOC). Ultimately, the correct derivative is derived using the FTOC and the Chain Rule, resulting in F'(x) = -sin(x)e^(cos^2(x)) - cos(x)e^(sin^2(x)). At x=0, this simplifies to F'(0) = -1. The thread concludes with a clarification on how to properly document the solution.
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Here is the question:

Derivative of integral?


F(x) = integral of e^(t^2)dt (upper limit = cosx, lower limit = sinx)
Now find F'(x) at x=0

HOW DO I SOLVE THIS :O

Okay so this is what I did,
solve integration, answer is [e^(t^2)]/2t
[2t is the derivative of power(t^2), you're suppose to DIVIDE the integration by the derivative, riiight?]

I put t=cosx, t=sinx
So, it becomes

e^({cosx}^2)]/2cosx - e^({sinx}^2)]/2cosx

Now I take its derivative...
which turns out to be very complicated so I think I'm doing it wrong, because it is supposed to be not-so-long.

THIS IS THE ANSWER GIVEN AT THE BACK:
Answer:
F ′(x)=exp (cos2 (x)) ·−sin (x)−exp (sin2 (x)) · cos (x) by FTOC
F ′(0)=exp (1) · 0−exp (0) · 1=−1

NOW HOLD ON A SECOND.
ISNT DERIVATIVE OF AN INTEGRAL, THE FUNCTION ITSELF? YESSSS.

OKAY, BUT THE DERIVATIVE AND INTEGRAL DONT UMMM CANCEL OUT TILL THE dx/dy/dt IS SAME WITH DERIVATIVE AND INTEGRATION!

Okay, I somehow solved the question B) *pat pat*
Can someone tell me how do i write this down on my paper?? O.o

I have posted a link there to this thread so the OP can view my work.
 
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Hello SnowPatrol,

We are given:

$$F(x)=\int_{\sin(x)}^{\cos(x)} e^{t^2}\,dt$$

And we are asked to find $F'(0)$.

By the FTOC and the Chain Rule, we have:

$$\frac{d}{dx}\int_{g(x)}^{h(x)} f(t)\,dt=f(h(x))\frac{dh}{dx}-f(g(x))\frac{dg}{dx}$$

Applying this formula, we find

$$F'(x)=-\sin(x)e^{\cos^2(x)}-\cos(x)e^{\sin^2(x)}$$

Hence:

$$F'(0)=-\sin(0)e^{\cos^2(0)}-\cos(0)e^{\sin^2(0)}=0-1=-1$$
 
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