Cal's questions at Yahoo Answers regarding the derivative form of the FTOC

[MarkFL]

Here are the questions:

How to get the derivative of a definite integral?

F(x) = the integral from x to 4 of sin(t^4) dt

F'(x) = ?

----

f(x) = the integral from x to 12 of t^7 dt

f'(x) = ?

I have several more like these. I've been working on them for 6 hours. None of my answers are right. I don't understand what I'm doing wrong!

I have posted a link there to this topic so the OP can see my work.

 

[MarkFL]

Hello Cal,

The derivative form of the Fundamental Theorem of Calculus states:

$$\frac{d}{dx}\int_a^{u(x)} f(t)\,dt=f\left(u(x) \right)\frac{du}{dx}$$

Another rule we will need is the following rule for definite integrals:

$$\int_a^b f(x)\,dx=-\int_b^a f(x)\,dx$$

This rule can easily be demonstrated using the anti-derivative form of the Fundamental Theorem of Calculus:

$$\int_a^b f(x)\,dx=F(b)-F(a)=-\left(F(a)-F(b) \right)=-\int_b^a f(x)\,dx$$

So, let's apply these rules to the given problems.

1.) $$F(x)=\int_x^4 \sin\left(t^4 \right)\,dt=-\int_4^x \sin\left(t^4 \right)\,dt$$

Hence:

$$F'(x)=-\frac{d}{dx}\int_4^x \sin\left(t^4 \right)\,dt=-\sin\left(x^4 \right)$$

2.) $$f(x)=\int_x^{12}t^7\,dt=-\int_{12}^x t^7\,dt$$

Hence:

$$f'(x)=-\frac{d}{dx}\int_{12}^x t^7\,dt=-x^7$$