Anyone can answer my question,

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In summary, the conversation discusses the relationship between the derivative and integral, specifically in the context of Leibniz and Newton's fundamental formula. The speaker also mentions the importance of understanding antiderivatives and their relationship to derivatives. The phrase "differentiating both sides as integrals of the upper limits" refers to the process of using the fundamental theorem of calculus to find the derivative of an integral. The speaker also clarifies that this is a special case of Leibniz's rule, but is commonly known as the fundamental theorem of calculus.
  • #1
TonyAlmeidaAtLSE
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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.
 
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  • #2
Consider the example:
[tex]f(x)=:\int_{a}^{x} u(t) dt [/tex] (1)
Take the primitive (antiderivative) of u(t) to be U(t).Then,equation (1) becomes,after applying the fundamental formula of Leibniz & Newton
[tex] f(x)=U(x)-U(a) [/tex] (2)
These 2 functions differ through a constant,namely U(a).So it's natural to conclude they have the same derivative.
[tex] f'(x)=U'(x)=u(x) [/tex] (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
[tex] \frac{d}{dx}\int_{a}^{x} u(t) dt=u(x) [/tex]
,where "a" is a constant.

Daniel.
 
  • #3
Although it is a special case of Leibniz rule, it is really just the "fundamental theorem of calculus".
 

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