What Does Differentiating Both Sides as Integrals of the Upper Limits Mean?

[TonyAlmeidaAtLSE]

Hi, I know I sound the most stupid here but I have an easy question here please help out:

X'(t)=dX(t)/dt

If dX/a(x) = b(t)dt
∫d(x)/a(x) = ∫b(s) ds {The upper and lower limites of lefthand side are X(t) and X(0) respectively and those of righthand side are t and 0}

Then differentiating both sides as integrals of the upper limits
X’(t)/a(X(t)) =b(t)
I don’t understand what does “differentiating both sides as integrals of the upper limits” mean? Here does it have anything to do with Leibniz Rule? What about lower limits X(0)

Your generous help and reply will be greatly appreciated.

 

[dextercioby]

Consider the example:
$$f(x)=:\int_{a}^{x} u(t) dt$$ (1)
Take the primitive (antiderivative) of u(t) to be U(t).Then,equation (1) becomes,after applying the fundamental formula of Leibniz & Newton
$$f(x)=U(x)-U(a)$$ (2)
These 2 functions differ through a constant,namely U(a).So it's natural to conclude they have the same derivative.
$$f'(x)=U'(x)=u(x)$$ (3)
,where i made use of the fact that U is the antiderivative of u,no matter whether the latter's argument is "x","t","v","y",...
Formula (3) means
$$\frac{d}{dx}\int_{a}^{x} u(t) dt=u(x)$$
,where "a" is a constant.

Daniel.

 

[HallsofIvy]

Although it is a special case of Leibniz rule, it is really just the "fundamental theorem of calculus".