経済学で出る数学

ワークブックでじっくり攻める：問 5.20

1. $f^{\prime}(x)=\dfrac{\left(x^2\right)^{\prime}\left(3-x\right)-\left(x^2\right)\left(3-x\right)^{\prime}}{\left(3-x\right)^2} =\dfrac{2x(3-x)-x^2\times(-1)}{\left(3-x\right)^2} =\dfrac{6x-2x^2+x^2}{\left(3-x\right)^2} =\dfrac{6x-x^2}{\left(3-x\right)^2} =\dfrac{x(6-x)}{\left(3-x\right)^2}$．


2. $f^{\prime}(x)=4+100\times\dfrac{-\left(x\right)^{\prime}}{\left(x\right)^2} =4-\dfrac{100}{x^2}=\dfrac{4x^2-100}{x^2} =\dfrac{4\left(x^2-25\right)}{x^2} =\dfrac{4(x+5)(x-5)}{x^2}$．


3. $f^{\prime}(x)=\dfrac{\left(x\right)^{\prime}\left(x+4\right)-\left(x\right)\left(x+4\right)^{\prime}}{\left(x+4\right)^2} =\dfrac{1\times\left(x+4\right)-\left(x\right)\times 1}{\left(x+4\right)^2} =\dfrac{4}{\left(x+4\right)^2}$．


4. $f^{\prime}(x)=\dfrac{\left(x^{\frac{1}{2}}\right)^{\prime}\left(x+1\right)-\left(x^{\frac{1}{2}}\right)\left(x+1\right)^{\prime}}{\left(x+1\right)^2} =\dfrac{\left(\dfrac{1}{2}x^{-\frac{1}{2}}\right)^{\prime}\left(x+1\right)-\left(x^{\frac{1}{2}}\right)\times 1}{\left(x+1\right)^2}\\ =\dfrac{\left(\dfrac{1}{2x^{\frac{1}{2}}}\right)\left(x+1\right)-\left(x^{\frac{1}{2}}\right)\times 1}{\left(x+1\right)^2} =\dfrac{\left(\dfrac{1}{2\sqrt{x}}\right)\left(x+1\right)-{\sqrt{x}}}{\left(x+1\right)^2}\\ =\dfrac{\left(\dfrac{1}{2\sqrt{x}}\right)\left(x+1-2x\right)}{\left(x+1\right)^2} =\dfrac{1-x}{2\sqrt{x}\left(x+1\right)^2} $．

$1$階条件を見越して，分子は可能な限り因数分解．分母の $(　)^2$ を，どーしても展開したがるが，それは無益．