【微分積分】3-2-2 対数微分法｜問題集

両辺の対数をとると、
\(\displaystyle \log y=2\log x+\frac{1}{2}\{\log(1+x^2)-\log(1-x^2)\}\)
両辺を\(x\)で微分すると、
\(\displaystyle \frac{y'}{y}=\frac{2}{x}+\frac{1}{2}\left\{\frac{2x}{1+x^2}-\frac{-2x}{1-x^2}\right\}\)
\(\displaystyle =\frac{2}{x}+\frac{1}{2}\frac{2x}{(1+x^2)(1-x^2)}\)
\(\displaystyle y'=y\left(\frac{2(1+x^2-x^4)}{x(1+x^2)(1-x^2)}\right)\)
\(\displaystyle =x^2\sqrt{\frac{1+x^2}{1-x^2}}\left(\frac{2(1+x^2-x^4)}{x(1+x^2)(1-x^2)}\right)\)
\(\displaystyle =\frac{2x(1+x^2-x^4)}{(1-x^2)\sqrt{1-x^4}}\)

両辺の対数をとると、
\(\log y=2\log x\)
\(\displaystyle \ \ \ \ \ +\frac{1}{2}\{\log(x^3+2x+1)-\log(x^2-3x+1)\}\)
両辺を\(x\)で微分すると、
\(\displaystyle \frac{y'}{y}=\frac{2}{x}+\frac{1}{2}\left(\frac{3x^2+2}{x^3+2x+1}-\frac{2x-3}{x^2-3x+1}\right)\)
\(\displaystyle y'=y\left\{\frac{2}{x}+\frac{1}{2}\left(\frac{3x^2+2}{x^3+2x+1}-\frac{2x-3}{x^2-3x+1}\right)\right\}\)
\(\displaystyle =x^2\sqrt{\frac{x^3+2x+1}{x^2-3x+1}}\)
\(\displaystyle \ \ \ \ \ \left\{\frac{2}{x}+\frac{1}{2}\left(\frac{3x^2+2}{x^3+2x+1}-\frac{2x-3}{x^2-3x+1}\right)\right\}\)

両辺の対数をとると、
\(\log y=\log x^x\)
\(\log y=x\log x\)
両辺を\(x\)で微分すると、
\(\displaystyle \frac{y'}{y}=\log x+x・\frac{1}{x}\)
\(\displaystyle y'=y\left(\log x+1\right)\)
\(\displaystyle =x^x(\log x+1)\)

両辺の対数をとると、
\(\log y=\log x^{\frac{1}{x}}\)
\(\displaystyle \log y=\frac{1}{x}\log x\)
両辺を\(x\)で微分すると、
\(\displaystyle \frac{y'}{y}=-\frac{1}{x^2}\log x+\frac{1}{x}・\frac{1}{x}\)
\(\displaystyle y'=y・\frac{1}{x^2}(1-\log x)\)
\(\displaystyle =x^{\frac{1}{x}-2}(1-\log x)\)