Help with Part 1 of the Fundamental Theorem of Calculus?? What does this mean? (d/dx) integral x to a f(t) dt = f(x)??????

Question

Help with Part 1 of the  Fundamental Theorem of Calculus?? What does this mean? (d/dx) integral x to a f(t) dt = f(x)??????

anonymous · Answer

it means, in english, that "the derivative of the integral is the integrand"

anonymous · Answer

I'm confused about the d/dx part in particular...

anonymous · Answer

that is, if 
$$F(x)=\int_a^xf(t)dt$$ then 
$$F'(x)=f(x)$$

anonymous · Answer

notice that 
$$F(x)=\int_a^xf(t)dt$$ is a function of the variable $x$ and not of $t$

anonymous · Answer

for example, if 
$$F(x)=\int_0^x\sin(t)dt$$ then 
$$F'(x)=\sin(x)$$

anonymous · Answer

So what does that mean d/dx mean exactly? Does it have todo with the dx dummy variable? Sorry, but that d/dx I throwing me off. Does it mean I have to take the derivative once I find the anti derivative?

anonymous · Answer

the $\frac{d}{dx}$ notation just means the derivative wrt $x$
do not be confused by that, it is the same as saying the derivative of the integral is the integrand