経済学で出る数学

ワークブックでじっくり攻める：問 5.21

1. $f^{\prime}(x) =\left(x\right)^{\prime}\left(e^x\right)+\left(x\right)\left(e^x\right)^{\prime} =1\times e^x+xe^x=(1+x)e^x$．


2. $f^{\prime}(x) =\left(x^{-1}\right)^{\prime}\left(e^x\right)+\left(x^{-1}\right)\left(e^x\right)^{\prime} =-x^{-2}e^x+x^{-1}e^x=x^{-2}\left(-e^x+xe^x\right) =\dfrac{e^x(x-1)}{x^2}$．


3. $\left(e^{3x}\right)^{\prime}=\left(e^x\times e^x\times e^x\right)^{\prime} =\left(e^{x}\right)^{\prime}\left(e^{x}\right)\left(e^{x}\right) +\left(e^{x}\right)\left(e^{x}\right)^{\prime}\left(e^{x}\right) +\left(e^{x}\right)\left(e^{x}\right)\left(e^{x}\right)^{\prime} =e^x\times e^x\times e^x+e^x\times e^x\times e^x+e^x\times e^x\times e^x =3e^{3x} $であることから，
   $f^{\prime}(x)=3e^3-3e^{3x}$．


4. $f(x)=3\log_{}{x}-3x$ なので， $f^{\prime}(x) =\dfrac{3}{x}-3=\dfrac{3(1-x)}{x}$．


5. $f^{\prime}(x) =\dfrac{\left(x^2\right)^{\prime}\left(e^x\right)-\left(x^2\right)\left(e^x\right)^{\prime}}{\left(e^x\right)^2} =\dfrac{2xe^x-x^2e^x}{\left(e^x\right)^2} =\dfrac{x(2-x)}{e^x}$．


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

$1$階条件を見越して，可能な限り因数分解．