I left out one line at the beginning, which was pointed out in class.
We start with some function, which is continuous on . We define:
By the fundamental theorem of calculus, this is continuous of and differentiable on .
Now, we use the mean value theorem for derivatives (somewhere in an interval, a function has a gradient which is equal to its average gradient) which say that if is differentiable on and continuous on then there exists some between and such that .
Now simply using our function in the mean value theorem for derivatives, we have that:
But we know that . We also have defined above, so we can plug in and to get:
The second term on the right hand side is zero as the two limits are the same, so we have:
But we know that as and are just dummy variables, and so we have proved that:
This finishes the proof.
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