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Could someone walk me through this derivative?


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The solution you linked corresponds to a simple case of Leibniz's Rule for Differentiating Integrals.

Note, however, that there is an obvious typo in that solution: there should be $ - \frac{1}{k}Qf(\frac{Q}{k})$ instead of $ - \frac{1}{k}Q (\frac{Q}{k})$.

EDIT: While the solution you linked is instructive, it is by far not the simplest one. In particular, note that $$ \frac{d}{{dQ}}\int_{Q/k}^\infty {Qf(x)dx} = \frac{d}{{dQ}}\bigg[Q\int_{Q/k}^\infty {f(x)dx} \bigg] = \bigg[1 - F\bigg(\frac{Q}{k}\bigg)\bigg] + Q\frac{d}{{dQ}}\bigg[1 - F\bigg(\frac{Q}{k}\bigg)\bigg]. $$

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