How do I find the derivative of this?
one of dem fukin tricks doe
wrt x function of f
>>8042706
what exactly are we trying to do here?
>>8042706
Volterra equ of 1st kind
>>8042706
expand f(t) with a mclauren series, solve integral, apply limit, rearrange sum back into closed form
What's going on here?
g is the integral of h(x) := x·f(x),
so the derivative of g is h.
>>8042706
https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus#First_part
>>8042706
Assuming the derivative is with respect to x. Otherwise this is a pointless problem.
Make t*f(t) = h(t). The antiderivative of h(t) is H(t). Run with it.
>>8042706
Let [math] \displaystyle h(t)=tf(t)[\math].
Then [math] \displaystyle \forall x\in\mathcal{R}, g(x)=\int_0^x h(t)dt[\math].
And by the fundamental theorem of calculus, [math] \displaystyle g'(x)=h(x)=xf(x) [\math].
If you need to go a step further, you could then say that [math] \displaystyle g''(x)=f(x)+xf'(x) [\math].
>>8043151
Sorry OP, wrong slash
Let [math] \displaystyle h(t)=tf(t)[/math].
Then [math] \displaystyle \forall x\in\mathcal{R}, g(x)=\int_0^x h(t)dt[/math].
And by the fundamental theorem of calculus, [math] \displaystyle g'(x)=h(x)=xf(x) [/math].
If you need to go a step further, you could then say that [math] \displaystyle g''(x)=f(x)+xf'(x) [/math].
>>8042706
OP Search up leibnez rule. It's solved to use volterra integrals by putting them into a differential equations form that you can solve via characteristic methods or other techniques.
>>8043157
Specifically this picture is what I wanted you to see.
Also, everybody in this thread is fucking retarded. They don't even know what they're talking about.
[math]\frac{d}{dx} \int_0^x t f(t) dt[/math]
definition of derivative:
[math]\lim_{h \to 0} \frac{\int_0^{x+h} t f(t) dt - \int_0^x t f(t) dt}{h}[/math]
Simplifies to:
[math]\lim_{h \to 0} \frac{\int_x^{x+h} t f(t) dt}{h}[/math]
The limit as you approach 0 leaves just the single term evaluated at x multiplied by the infinitesimal and it cancels out with the infinitesimal in the denominator leaving you with:
[math]xf(x)[/math]
