経済学で出る数学

ワークブックでじっくり攻める：問 5.22

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


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


3. $f^{\prime}(x) =e^{2x}\times 2=2e^{2x}$．


4. $f^{\prime}(x) =e^{3x^2}\times \left(6x\right)=6xe^{3x^2}$．