# Homework Help: Rieman Integral: The Fundamental Theorem of Calculus

1. Apr 21, 2012

### kingstrick

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

Let I := [a,b] and let f: I→ℝ be continuous on I. Also let J := [c,d] and let u: J→ℝ be differentiable on J and satisfy u(J) contained in I. Show that if G: J→ℝ is defined by

G(x) :=∫u(x)af for x in J, then G'(x) = (f o u)(x)u'(x) for all x in J.

2. The attempt at a solution

I am not sure what theorem to use in this situation. I am thinking that the product and composition of differentiable functions means that G'(x) exists....but I don't believe that will help me in this situation. Any ideas as to how i should start this one?

2. Apr 21, 2012

### Dick

3. Apr 21, 2012

### HallsofIvy

The Leibniz formula that Dick links to is basically using the chain rule.

If $F(x)=\int_a^{g(x)} f(t)dt$ let u= g(x) so that the integral is
$\int_a^u f(t)dt$ and, by the Fundamental theorem of Calculus,
$$\frac{d}{du}\int_a^u f(t)dt= f(u)$$

Now, use the chain rule to find the derivative with respect to x.

4. Apr 22, 2012

### kingstrick

I don't understand. Isn't the derivative of G(x) the f in the integral. Or am i supposed to find the derivative of F(X)?

5. Apr 22, 2012

### HallsofIvy

No, the derivative of G(x) is NOT "the f in the integral". If the upper limit were just "x" it would be.