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Let us say we have a function defined by a definite integral, and that the function depends on some parameter, *p*. From the definition of an Riemann Integral we have

Let us now take the derivative of this with respect to *p*, what would we get? (We shall here approach the problem in a somewhat semi-formal way. For a more formal proof you may google for Leibniz integral rule.) We get

or

So

The last step was done because since *h* is a constant, as seen from the inner limit, we may put it inside the sum (this has to be proven – but for now I’ll let it pass).

Now, since *h* and Δ*x* are independent, we may switch the limits, to get

This is yet another step that may need some more rigorous work – but it will do for now. In this we recognize the definition of derivative, and we may thus write

or

Ok, so we may switch the order – how would that help us? I mean, we don’t get the value of the integral – right?

In the next few sections, we will look at some examples of how this could be used.

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