Anybody know how to differentiate:
Int[0,x](f(k)(x-k))dk
with respect to x?
No worries:
Differentiation under the integral sign - Wikipedia, the free encyclopedia
I may be the first person to get a degree using nothing but wikipedia...
Wikipedia is usually over my head I usually use "brute force" where I try every possible method until I eventually arrive at a correct answer, then I try that method on the next question and see if it works :P
On the down side, it probably took me 80 hours to figure out how to differentiate. >.<
In your case the borns of the integral are 0 and x. Thefore, for a consequence of some theorem (that I don't remember exactly), the derivative of your integral is equal to what is under the integral. Notice that your integral is dt, not dx...
Imagine you had the same integral, but with the upper born equal to $\displaystyle x^2$, then differentiate your integral, respect to x would be $\displaystyle f(k)(x^2-k)(2x)$, it's a consequence of the chain rule I think.
Ok I'm probably wrong, but reading my class notes on calculus, I found :
If h is continue, f and g derivable and $\displaystyle F(x)=$ Integral from f(x) to g(x) of h(t) dt, then $\displaystyle F'(x)=h(g(x))g'(x)-h(f(x))f'(x)$.
With this formula, you can calcul the derivative of your problem. And this formula is true.
Hope this helps, at least for further derivatives of integrals involving 2 borns in function of x.
I know that and you know that. But if incorrect posts are made, they have to be addressed. Otherwise others who read this thread might be mislead. There is clearly some confusion on:
1. The significnace of the integrand containing the variable of differentiation.
2. When the FVC can be applied.
This confusion has to be pointed out and addressed.