1.次の関数を微分しなさい。
(1)\(y=\log3x\)
\(\displaystyle y'=\frac{(3x)'}{3x}\)
\(\displaystyle =\frac{1}{x}\)
(2)\(y=\log_{2}(4x-1)\)
\(\displaystyle y'=\frac{(4x-1)'}{(4x-1)\log2}\)
\(\displaystyle =\frac{4}{(4x-1)\log2}\)
(3)\(y=\log(x^2+1)\)
\(\displaystyle y'=\frac{(x^2+1)'}{x^2+1}\)
\(\displaystyle =\frac{2x}{x^2+1}\)
(4)\(y=x\log x-x\)
\(y'=\log x+1-1\)
\(=\log x\)
(5)\(y=\log|3x+2|\)
\(\displaystyle y'=\frac{(3x+2)'}{3x+2}\)
\(\displaystyle =\frac{3}{3x+2}\)
(6)\(y=\log|\log x|\)
\(\displaystyle y'=\frac{(\log x)'}{\log x}\)
\(\displaystyle =\frac{1}{x\log x}\)
(7)\(y=\log|\sin x|\)
\(\displaystyle y'=\frac{(\sin x)'}{\sin x}\)
\(\displaystyle =\frac{\cos x}{\sin x}\)
(8)\(y=\log_{2}|x^2-4|\)
\(\displaystyle y'=\frac{(x^2-4)'}{(x^2-4)\log2}\)
\(\displaystyle =\frac{2x}{(x^2-4)\log2}\)
(9)\(\displaystyle y=\log\left|\frac{x+1}{x+2}\right|\)
\(y=\log|x+1|-\log|x+2|\)
\(\displaystyle y'=\frac{1}{x+1}-\frac{1}{x+2}\)
\(\displaystyle =\frac{1}{(x+1)(x+2)}\)
(10)\(\displaystyle y=\frac{(x+3)^2}{(x-1)(2x-1)}\)
両辺の絶対値の自然数をとると、
\(\displaystyle \log|y|=\log\left|\frac{(x+3)^2}{(x-1)(2x-1)}\right|\)
\(\displaystyle \log|y|=2\log|x+3|-\log|x-1|-\log|2x-1|\)
両辺を\(x\)で微分すると、
\(\displaystyle \frac{y'}{y}=2・\frac{(x+3)'}{x+3}-\frac{(x-1)'}{x-1}-\frac{(2x-1)'}{2x-1}\)
\(\displaystyle \frac{y'}{y}=\frac{2}{x+3}-\frac{1}{x-1}-\frac{2}{2x-1}\)
\(\displaystyle \frac{y'}{y}=\frac{-15x+11}{(x+3)(x-1)(2x-1)}\)
よって、
\(\displaystyle y'=\frac{-15x+11}{(x+3)(x-1)(2x-1)}・\frac{(x+3)^2}{(x-1)(2x-1)}\)
\(\displaystyle =-\frac{(x+3)(15x-11)}{(x-1)^2(2x-1)^2}\)
(11)\(y=x^2\sqrt[3]{x+1}\)
両辺の絶対値の自然数をとると、
\(\displaystyle \log|y|=2\log|x|+\frac{1}{3}\log|x+1|\)
両辺を\(x\)で微分すると、
\(\displaystyle \frac{y'}{y}=2・\frac{(x)'}{x}+\frac{1}{3}・\frac{(x+1)'}{x+1}\)
\(\displaystyle \frac{y'}{y}=\frac{2}{x}+\frac{1}{3(x+1)}\)
\(\displaystyle \frac{y'}{y}=\frac{7x+6}{3x(x+1)}\)
よって、
\(\displaystyle y'=\frac{7x+6}{3x(x+1)}・x^2\sqrt[3]{x+1}\)
\(\displaystyle =\frac{x(7x+6)}{3\sqrt[3]{(x+1)^2}}\)
(12)\(\displaystyle y=\frac{(x+2)(x+3)^3}{x^2+1}\)
両辺の絶対値の自然数をとると、
\(\displaystyle \log|y|=\log|x+2|+3\log|x+3|-\log|x^2+1|\)
両辺を\(x\)で微分すると、
\(\displaystyle \frac{y'}{y}=\frac{(x+2)'}{x+2}+3・\frac{(x+3)'}{x+3}-\frac{(x^2+1)'}{x^2+1}\)
\(\displaystyle \frac{y'}{y}=\frac{1}{x+2}+\frac{3}{x+3}-\frac{2x}{x^2+1}\)
\(\displaystyle \frac{y'}{y}=\frac{2x^3-x^2-8x+9}{(x+2)(x+3)(x^2+1)}\)
よって、
\(\displaystyle y'=\frac{2x^3-x^2-8x+9}{(x+2)(x+3)(x^2+1)}・\frac{(x+2)(x+3)^3}{x^2+1}\)
\(\displaystyle =\frac{(x+3)^2(2x^3-x^2-8x+9)}{(x^2+1)^2}\)
(13)\(y=x^{\sin x}\)
両辺の絶対値の自然数をとると、
\(\displaystyle \log|y|=\sin x\log|x|\)
両辺を\(x\)で微分すると、
\(\displaystyle \frac{y'}{y}=(\sin x)'\log|x|+x(\log|x|)'\)
\(\displaystyle \frac{y'}{y}=\cos x\log x+\frac{\sin x}{x}\)
よって、
\(\displaystyle y'=\left(\cos x\log x+\frac{\sin x}{x}\right)x^{\sin x}\)
(14)\(y=e^{2x}\)
\(y'=e^{2x}・(2x)'\)
\(=2e^{2x}\)
(15)\(y=e^{-x^2}\)
\(y'=e^{-x^2}・(-x^2)'\)
\(=-2xe^{-x^2}\)
(16)\(y=3^x\)
\(y'=3^x\log3\)
(17)\(y=2^{-3x}\)
\(y'=2^{-3x}\log2・(-3x)'\)
\(=-3・2^{-3x}\log2\)
(18)\(y=xe^x\)
\(y'=(x)'e^{x}+x(e^x)'\)
\(=e^x(1+x)\)
(19)\(y=(2x-1)a^x\)
\(y'=(2x-1)'a^{x}+(2x-1)(a^x)'\)
\(=2a^{x}+(2x-1)a^x\log a\)
\(=a^x\{2+(2x-1)\log a\}\)
(20)\(y=3^{3x-1}\)
\(y'=(3x-1)'・3^{3x-1}\log3\)
\(=3^{3x}\log3\)
(21)\(\displaystyle y=\frac{e^x}{e^x+1}\)
\(\displaystyle y'=\frac{e^x(e^x+1-e^xe^x)}{(e^x+1)^2}\)
\(\displaystyle =\frac{1}{(e^x+1)^2}\)
(22)\(\displaystyle y=\frac{e^x-e^{-x}}{e^x+e^{-x}}\)
\(y'\)
\(\displaystyle =\frac{\{e^x-(-e^{-x})\}(e^x+e^{-x})-(e^x-e^{-x})(e^x-e^{-x})}{(e^x+e^{-x})^2}\)
\(\displaystyle =\frac{4}{(e^x+e^{-x})^2}\)
