Derivative of Integral of Continuous Function f(x)

[kidia]

If f is continuous over an interval containing (a,x)find from first principles the derivative of the function f(x)=integral f(t)dt.Any help?

 

[Hessam]

dump rule

integral from a->x of f(t)dt = f(x) * d/dx(x) = f(x)

you can derive this easily
let F(x) be the antiderivative of f(x)

therefore the integral = F(x) - F(a)

take the derivative of that... the F(a) term falls off cause its a constant

thus you do d/dx(F(x)) = f(x)

yay!

if this is for AP calc, i would really recommend looking over the fundamental theorom of calculus

 

[HallsofIvy]

I don't think that's as "first principles" as kidia intended. Here is the standard proof of the fundamental theorem:
Let $$F(x)= \int_a^x f(t)dt$$. Then $$F(x+h)= \int_a^{x+h}f(t)dt$$
$$= \int_a^x f(t)dt+ \int_x^{x+h}f(t)dt$$

So that F(x+h)- F(x)= \int_x^{x+h}f(t)dt. Now apply the mean value theorem to the function $$\int_x^{x+h}f(t)dt$$ to argue that F(x+h)-F(x)= hf(x^*) where x^* is between x and x+h. Finally, divide both sides by h and take the limit as h goes to 0.