【高校数学Ⅲ】3-1-5 対数関数と指数関数の導関数｜要点まとめ

\(y'=2\log x・(\log x)'\)
\(\displaystyle =\frac{2\log x}{x}\)

\(\displaystyle y'=\frac{(3x+2)'}{(3x+2)\log2}\)
\(\displaystyle =\frac{3}{(3x+2)\log2}\)

\(\displaystyle y'=\frac{(x^2-3)'}{x^2-3}\)
\(\displaystyle =\frac{2x}{x^2-3}\)

両辺の絶対値の自然数をとると、
\(\displaystyle \log|y|=\log\left|\frac{\sqrt{2x-1}}{x-1}\right|\)
\(\displaystyle \log|y|=\frac{1}{2}\log|2x-1|-\log|x-1|\)
両辺を\(x\)で微分すると、
\(\displaystyle \frac{y'}{y}=\frac{1}{2}・\frac{(2x-1)'}{2x-1}-\frac{(x-1)'}{x-1}\)
\(\displaystyle \frac{y'}{y}=\frac{1}{2x-1}-\frac{1}{x-1}\)
\(\displaystyle \frac{y'}{y}=-\frac{x}{(2x-1)(x-1)}\)
よって、
\(\displaystyle y'=-\frac{x}{(2x-1)(x-1)}・\frac{\sqrt{2x-1}}{x-1}\)
\(\displaystyle =-\frac{x}{(x-1)^2\sqrt{2x-1}}\)

\(y'=(x^2)'e^{2x}+x^2(e^{2x})'\)
\(=2xe^{2x}+2x^2e^{2x}\)
\(=2x(1+x)e^{2x}\)

\(y'=2^{-x^2}・\log2・(-x^2)'\)
\(=-2x・2^{-x^2}\log2\)
\(=-2^{1-x^2}x\log2\)