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$$\frac{d}{dq}\int_{s_{1}-z-q}^{z-s_{1}} \varphi(w) \, dw$$

(if it helps, in my setting $\varphi$ is the CDF of some arbitrary uniform distribution). So I want to get a nice expression for this integral and it seems to suggest FTC, but I tried a change of variable and ended up with a $q$ inside the integrand which was not nice. Any help much appreciated.

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up vote 1 down vote accepted

Use the chain rule:

$$\frac{d}{dq}\int_{s_{1}-z-q}^{z-s_{1}} \varphi(w) \, dw = -\frac{d}{dq}\int_{z-s_1}^{s_1-z-q} \varphi(w) \, dw$$

$$ = - \frac{d}{d(s_1-z-q)} \int_{z-s_1}^{s_1-z-q} \varphi(w) \, dw \cdot\frac{d}{dq}(s_1-z-q)$$

$$ -\varphi(s_1-z-q)\cdot (-1). $$

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thanks! much appreciated – Red Rover Apr 22 '12 at 19:00

Use the Leibnitz Rule of Differentiation of Integrals in the case that the limits of integration depend on the differentiation variable.

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The Leibniz rule as stated in the linked article has the bounds of integration depending on the variable with respect to which one differentiates, but also the function being integrated depends on that variable. So it seems like a more powerful tool than what is needed in this case. – Michael Hardy Apr 22 '12 at 19:05
Though perhaps too powerful, it does indeed solve the problem. Sometimes one cares less about the mode of transportation than the destination.... ;) Plus, it isn't clear that $w$ isn't a function of $q$ from the OP. – user02138 Apr 22 '12 at 19:24

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