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\frac{\mathrm{d}}{\mathrm{dx}} \mathrm{A}(x)=\lim _{h \rightarrow 0} \frac{A(x+h)-A(x)}{h}=\lim _{h \rightarrow 0} \frac{1}{h}\left\{\int_{a}^{x+h} \mathrm{f}(\mathrm{t}) \mathrm{dt}-\int_{a}^{x} \mathrm{f}(\mathrm{t}) \mathrm{dt}\right\}=\lim _{h \rightarrow 0} \frac{1}{h} \int_{x}^{x+h} \mathrm{f}(\mathrm{t}) \mathrm{dt}