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\begin{align*} \begin{aligned} &\operatorname{D}\left(\frac{\mathrm{f}(x)}{g(x)}\right)=\lim _{h \rightarrow 0} \frac{1}{h}\left\{\frac{f(x+h)}{g(x+h)}-\frac{f(x)}{g(x)}\right\}=\lim _{h \rightarrow 0} \frac{1}{h}\left\{\frac{f(x+h) g(x)-g(x+h) f(x)}{g(x+h) g(x)}\right\} \\ &=\lim _{h \rightarrow 0} \frac{1}{g(x+h) g(x)}\left\{\frac{f(x+h) g(x)+(-f(x) g(x)+f(x) g(x))-g(x+h) f(x)}{h}\right\} \\ &=\lim _{h \rightarrow 0} \frac{1}{g(x+h) g(x)}\left\{g(x) \frac{f(x+h)-f(x)}{h}-f(x) \frac{g(x+h)-g(x)}{h}\right\} \\ &=\frac{1}{g^{2}(x)}\left\{g(x) \cdot f^{\prime}(x)-f(x) \cdot g^{\prime}(x)\right\}=\frac{g(x) \cdot f^{\prime}(x)-f(x) \cdot g^{\prime}(x)}{g^{2}(x)} \end{aligned} \end{align*}