【微分積分】3-2-2 対数微分法｜要点まとめ

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

両辺の対数をとると、
\(\displaystyle \log y=\log\sqrt{\frac{(x+1)^3}{(x-2)^2(x+3)^5}}\)
\(\displaystyle \log y=\frac{1}{2}(3\log|x+1|\)
\(\ \ \ \ \ \ -2\log|x-2|-5\log|x+3|)\)
両辺を\(x\)で微分すると、
\(\displaystyle \frac{y'}{y}=\frac{1}{2}\left(\frac{3}{x+1}-\frac{2}{x-2}-\frac{5}{x+3}\right)\)
\(\displaystyle =\frac{-2x^2-7}{(x+1)(x-2)(x+3)}\)
\(\displaystyle y'=y\frac{-2x^2-7}{(x+1)(x-2)(x+3)}\)
\(\displaystyle =\frac{-2x^2-7}{(x+1)(x-2)(x+3)}\sqrt{\frac{(x+1)^3}{(x-2)^2(x+3)^5}}\)

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