【微分積分】3-3-1 導関数の計算｜要点まとめ

\(y'=12(3x^5-2x^3+6)^{11}(15x^4-6x^2)\)
\(=36x^2(5x^2-2)(3x^5-2x^3+6)^{11}\)

\(y'=12(2x+5)^5(3x-4)^5\)
\(\ \ \ \ \ +(2x+5)^6・15(3x-4)^4\)
\(=(66x+27)(2x+5)^5(3x-4)^4\)

\(\displaystyle y'=\frac{1}{2\sqrt{x^2+1}}・2x\)
\(\displaystyle =\frac{x}{\sqrt{x^2+1}}\)

\(\displaystyle y'=\frac{\sqrt{1-x^2}-x・\frac{1}{2\sqrt{1-x^2}}・(-2x)}{1-x^2}\)
\(\displaystyle =\frac{\sqrt{1-x^2}+\frac{x^2}{\sqrt{1-x^2}}}{1-x^2}\)
\(\displaystyle =\frac{1}{(1-x^2)^{\frac{3}{2}}}\)

\(\displaystyle y'=\frac{4}{3}x^{\frac{1}{3}}\)
\(\displaystyle =\frac{4\sqrt[3]{x}}{3}\)

\(y=x^2+x+1+x^{-1}+x^{-2}\)
\(y'=2x+1-x^{-2}-2x^{-3}\)
\(\displaystyle =2x+1-\frac{1}{x^2}-\frac{2}{x^3}\)

\(y'=2\sin3x・(\sin3x)'\)
\(=2\sin3x・\cos3x・(3x)'\)
\(=6\sin3x\cos3x\)
\(=3\sin6x\)

\(y'=-\sin(x^2+1)・(x^2+1)'\)
\(=-2x\sin(x^2+1)\)

\(y'=3^{5x-7}(\log3)・5\)
\(=5・3^{5x-7}\log3\)

\(\displaystyle y'=\frac{\frac{1}{x}・x-\log x}{x^2}\)
\(\displaystyle =\frac{1-\log x}{x^2}\)

\(\displaystyle y'=\frac{1}{\log x}・(\log x)'\)
\(\displaystyle =\frac{1}{x\log x}\)

\(\displaystyle y'=(\log x)^2+x・2\log x・\frac{1}{x}\)
\(=(\log x)^2+2\log x\)

\(y'=3\tan^24x・(\tan4x)'\)
\(\displaystyle =3\tan^24x・\frac{1}{\cos^24x}・4\)
\(\displaystyle =\frac{12\tan^24x}{\cos^24x}\)

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

\(\displaystyle y'=\frac{1}{x+\sqrt{x^2+1}}\left(1+\frac{1}{2\sqrt{x^2+1}}・2x\right)\)
\(\displaystyle =\frac{1}{x+\sqrt{x^2+1}}・\frac{\sqrt{x^2+1}+x}{\sqrt{x^2+1}}\)
\(\displaystyle =\frac{1}{\sqrt{x^2+1}}\)

\(y'=2e^{2x}\cos3x-e^{2x}\sin3x・3\)
\(=e^{2x}(2\cos3x-3\sin3x)\)

\(\displaystyle y=\frac{1}{2}\log(1-\sin x)-\frac{1}{2}\log(1+\cos x)\)
\(\displaystyle y'=\frac{-\cos x}{2(1-\sin x)}-\frac{-\sin x}{2(1+\cos x)}\)
\(\displaystyle =\frac{\sin x-\cos x-1}{2(1-\sin x)(1+\cos x)}\)

\(\displaystyle y=\frac{1}{\sqrt{2}}\log|\sqrt{1-x}-\sqrt{2})\)
\(\displaystyle \ \ \ \ \ -\frac{1}{2}\log(\sqrt{1-x}+\sqrt{2})\)
\(\displaystyle y'=-\frac{1}{2\sqrt{2-2x}}\)
\(\displaystyle \ \ \ \ \ ・\left(\frac{1}{\sqrt{1-x}-\sqrt{2}}-\frac{1}{\sqrt{1-x}+\sqrt{2}}\right)\)
\(\displaystyle =\frac{1}{(x+1)\sqrt{1-x}}\)

\(\displaystyle y'=\frac{1}{\sqrt{1-(x^2)^2}}・(x^2)'\)
\(\displaystyle =\frac{2x}{\sqrt{1-x^4}}\)

\(\displaystyle y'=-\frac{1}{\sqrt{1-(e^{-x})^2}}・(e^{-x})'\)
\(\displaystyle =\frac{e^{-x}}{\sqrt{1-e^{-2x}}}\)
\(\displaystyle =\frac{1}{\sqrt{e^{2x}-1}}\)

\(\displaystyle y'=\sin^{-1}x+x・\frac{1}{\sqrt{1-x^2}}\)
\(\displaystyle \ \ \ \ \ +\frac{1}{2\sqrt{1-x^2}}・(-2x)\)
\(\displaystyle =\sin^{-1}x+\frac{x}{\sqrt{1-x^2}}-\frac{x}{\sqrt{1-x^2}}\)
\(\displaystyle =\sin^{-1}x\)

\(\displaystyle y'=\frac{1}{1+\left(\frac{x}{a}\right)^2}・\frac{1}{a}\)
\(\displaystyle =\frac{a}{a^2+x^2}\)

\(\displaystyle y'=\frac{1}{1+\left(\frac{a}{x}\right)^2}・\left(-\frac{a}{x^2}\right)\)
\(\displaystyle =-\frac{a}{a^2+x^2}\)

\(\displaystyle y'=-\frac{1}{\sqrt{1-\frac{x+1}{2}}}・\frac{1}{2\sqrt{\frac{x+1}{2}}}・\frac{1}{2}\)
\(\displaystyle =-\frac{1}{\sqrt{\frac{1-x}{2}}}・\frac{1}{2\sqrt{2x+2}}\)
\(\displaystyle =-\frac{1}{2\sqrt{1-x^2}}\)

\(\displaystyle y'=\frac{1}{1+(\frac{1-x}{1+x})^2}・\frac{-(1+x)-(1-x)}{(1+x)^2}\)
\(\displaystyle =\frac{1}{(1+x)^2+(1-x)^2}・(-2)\)
\(\displaystyle =-\frac{1}{1+x^2}\)

\(\displaystyle y=\frac{x\sin^{-1}x}{\sqrt{1-x^2}}+\frac{1}{2}\log(1-x^2)\)
\(\displaystyle y'=\frac{\sin^{-1}x}{(1-x^2)^{\frac{3}{2}}}+\frac{x}{1-x^2}-\frac{x}{1-x^2}\)
\(\displaystyle =\frac{\sin^{-1}x}{(1-x^2)^{\frac{3}{2}}}\)

\(\displaystyle y'=\cos(\cos^{-1}x^4)・\frac{-1}{\sqrt{1-(x^4)^2}}・4x^3\)
\(\displaystyle =x^4・\frac{-4x^3}{\sqrt{1-x^8}}\)
\(\displaystyle =-\frac{4x^7}{\sqrt{1-x^8}}\)

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

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

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