# A very quick question about definite integrals

1. Mar 10, 2010

### michonamona

1. The problem statement, all variables and given/known data

F(x) = $$\int^{x}_{0}f(t)dt$$

Then F'(x) = f(x)

what is f'(x)? is this equivalent to f(t)?

Thanks for your help
M

2. Mar 10, 2010

### Dick

F(x) is your integral of f from 0 to x. F'(x) is the derivative of F(x), which is f(x), the value of your integrand f(t) evaluated at t=x. This is just the fundamental theorem of calculus, that the integral is the antiderivative of the integrand.

3. Mar 10, 2010

### michonamona

Thanks for the reply.

so f '(x) IS indeed f(t)? the very same f(t) in F(x)?

4. Mar 10, 2010

### Dick

No. F'(x) is f(x). But, yes, the derivative of the integral is the function you are integrating, isn't that what the fundamental theorem of calculus is all about?

5. Mar 10, 2010

### michonamona

I'm sorry, I don't think my notations are clear. I understand that big F'(x) = f(x), what I'm concerned with is whether small f '(x) is f(t).

so can we write

F(x) = $$\int^{x}_{0}f'(x)dt$$ = $$\int^{x}_{0}f(t)dt$$

6. Mar 10, 2010

### Staff: Mentor

You don't have enough information to determine f'(x). The only information you have is that F is an antiderivative of f. A nearly equivalent way to say this is that f is the derivative of F. IOW, F'(x) = f(x).

For example, if F(x) = x3, F'(x) = f(x) = 3x2. To go a step further and find f'(x), you need to know what the function f(x) is.

7. Mar 10, 2010

### Dick

How can f'(x) be the same as f(t)? They don't even involve the same variable.

8. Mar 10, 2010

### michonamona

Thanks guys, it now makes sense. I keep getting all the notations mixed up.

I appreciate all of your help

M

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