In calculus the Leibniz integral rule for differentiation under the integral sign is based on the following theorem.
Theorem for differentiation under the integral sign:
Suppose a≤b and c≤d are four real constants.
Suppose also that f(x,y):P→R is a function which is continuous and has a continuous partial derivative fy(x,y) in the rectangle
P={(x,y)|a≤x≤b,c≤y≤d}
Suppose also that the functions
α(y),β(y):[c,d]→[a,b]
are differentiable in the interval [c,d]. Then the integral
Φ(y)=β(y)∫α(y)f(x,y) dx
(which is a function of y∈[c,d]) is a differentiable function of y in the interval [c,d] and
Φ′(y)=f(β(y),y)⋅β′(y)−f(α(y),y)⋅α′(y)+β(y)∫α(y)f′y(x,y) dx
Note:
In the special case when α(y)=α and β(y)=β are constant functions equality (2) takes the form
Φ′(y)=β∫αf′y(x,y) dx
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