経済学で出る数学

ワークブックでじっくり攻める：問 5.19

1. $f^{\prime}(x)= \left(x^2\right)^{\prime}\left(x^3+1\right)+\left(x^2\right)\left(x^3+1\right)^{\prime} =2x\left(x^3+1\right)+\left(x^2\right)\left(3x^2\right) =2x^4+2x+3x^4=5x^4+2x$．


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


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


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

$1$階条件を見越すと，微分した結果が因数分解されていると好都合．

1. $5x^4+2x=x(5x^3+2)$・・・もうちょっと因数分解できるが省略f(^-^;


2. $4x^3-2x=2x\left(2x^2-1\right)=2x(\sqrt{2}x+1)(\sqrt{2}x-1)$


3. $3x^2+6x=3x(x+2)$


4. $3x^2-2x-1=(3x+1)(x-1)$