Differentiation under the integral sign is a useful operation in calculus. Suppose that it is required to differentiate with respect to x the function

F(x)= int_{a(x)}^{b(x)}f(x,t) ,dt,

where the functions f(x,t) , and frac{ partial}{ partial x} ,f(x,t) , are both continuous in both t , and x , in some region of the (t,x) , plane, including a(x) leq t leq b(x) ,, x_0 leq x leq x_1 ,, and the functions a(x) , and b(x) , are both continuous and both have continuous derivatives for x_0 leq x leq x_1 ,. Then for ,x_0 leq x leq x_1 , ,:

frac{d}{dx} ,F(x) = f(x,b(x)) ,b'(x) - f(x,a(x)) ,a'(x) + int_{a(x)}^{b(x)} frac{ partial}{ partial x} , f(x,t) ; dt ,.

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