The differentiation of definite integrals

I must now make a digression. I want to show how to find the differential coefficient, du/da, of the definite integral u = Jf x, a)dx between the limits yx and y0, when yx and y0 are functions of a. Let /(a , a)dx become f(x, a) after integration, we have therefore 

on partial differentiation with respect to yv when y0 is constant and then with respect to y0, when y1 is constant, we get d 7)u d 

Now suppose that a suffers a small increment so that when a becomes a + h, u becomes u + k, then, keeping the limits constant, Incr. v= f f (x, a + h)-f(x, a) dx.. . (69) 

If both yx and y0 are functions of a, then du/da must be the sum of three separate terms, (i) the change due to a (ii) the change due to yx and (iii) the change due to y0. These separate effects have been evaluated in equations (68) and (70), consequently, 

The higher derivatives can be obtained by an application of the same methods. 

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