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# differentiating the integral watch

1. hello,

so i know that if i differentiate w.r.t. x {integral of f(x) dx} i get f(x)

but what do i get

if i differentiate w.r.t. x {integral from a to b of f(x) dx}

or

if i differentiate w.r.t. x {integral from v(x) to u(x) of f(x) dx}

?????????????
2. Are you talking about the differentiation involved in C3?
3. no, i'm just self studying, reading a book from Mary L. Boas
4. (Original post by pepsigirl)
hello,

so i know that if i differentiate w.r.t. x {integral of f(x) dx} i get f(x)

but what do i get

if i differentiate w.r.t. x {integral from a to b of f(x) dx}
0. The integral is just a constant, after all.

if i differentiate w.r.t. x {integral from v(x) to u(x) of f(x) dx}
Suppose . Then F'(x) = f(x). Now, use the chain rule to solve your actual problem.
5. (Original post by pepsigirl)
hello,

so i know that if i differentiate w.r.t. x {integral of f(x) dx} i get f(x)

but what do i get

if i differentiate w.r.t. x {integral from a to b of f(x) dx}

or

if i differentiate w.r.t. x {integral from v(x) to u(x) of f(x) dx}

?????????????
For your first question, the integral from a to b of f(x) dx would be a constant. Therefore, the derivative of that would simply be zero.

f(u(x))*u'(x) - f(v(x))*v'(x)

And also, if you're given a mixed type of problem (with a constant and a function), the same rules apply.

For example, differentiate w.r.t. x {integral from A to U(x) of f(x) dx} would be 0 - f(U(x))*U'(x)
6. i 've looked up this rule called Leibniz's rule it says

d/dx of integral from u(x) to v(x) of f(x,t) dt =

f(x,v) (dv/dx) - f(x,u) (du/dx)
+integral from u(x) to v(x) of partial derivative of f w.r.t. x dt

and so with regard your answer i don't understand this last bit, b/c in the case of

d/dx of integral from u(x) to v(x) of f(x) dx =

f(v) v'(x) - f(u) u'(x) PLUS
integral from u(x) to v() of partial deriv. of f(x) w.r.t x dx

?????
7. it must be something really obvious and i just can't see it i know... but i cant
8. (Original post by pepsigirl)
i 've looked up this rule called Leibniz's rule it says

d/dx of integral from u(x) to v(x) of f(x,t) dt =

f(x,v) (dv/dx) - f(x,u) (du/dx)
+integral from u(x) to v(x) of partial derivative of f w.r.t. x dt

and so with regard your answer i don't understand this last bit, b/c in the case of

d/dx of integral from u(x) to v(x) of f(x) dx =

f(v) v'(x) - f(u) u'(x) PLUS
integral from u(x) to v() of partial deriv. of f(x) w.r.t x dx

?????
I can't that I know anything about Leibniz's rule; however, from looking at your example and from the wikipedia article, I think that in our cases, discussed above, f(x,t) is only a function of one variable (f(x) or f(t) depending on whether we have dx or dt).

In that case, I believe that makes the second part of the equation "the PLUS" part tend to zero and thus can be ignored.

EDIT: Looking at it more, I think that if you define f(x,t) as a function of one variable, then the last part of it - that whole complicated partial part - would be a function of 'x' or 't' while the partial derivative would be of the opposite value, 't' or 'x' respectively, and therefore would be zero - which means the integral would be zero (regardless of the limits).

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