【微分積分】3-2-1 初等関数の微分公式｜問題集
1.次の関数の導関数を求めなさい。
(1)\(y=x^2e^x\)
\(y'=2xe^x+x^2e^x\)
\(=x(x+2)e^x\)
\(=x(x+2)e^x\)
(2)\(y=e^x\sin x\)
\(y'=e^x\sin x+e^x\cos x\)
\(=e^x(\sin x+\cos x)\)
\(=e^x(\sin x+\cos x)\)
(3)\(\displaystyle y=\frac{e^x}{\sin x}\)
\(\displaystyle y'=\frac{e^x\sin x-e^x\cos x}{\sin^2x}\)
\(\displaystyle =\frac{e^x(\sin x-\cos x)}{\sin^2x}\)
\(\displaystyle =\frac{e^x(\sin x-\cos x)}{\sin^2x}\)
(4)\(y=x^2\log x\)
\(\displaystyle y'=2x\log x+x^2・\frac{1}{x}\)
\(\displaystyle =2x\log x+x\)
\(\displaystyle =2x\log x+x\)
(5)\(y=x^3\sin2x\)
\(y'=3x^2\sin2x+x^3・2\cos2x\)
\(=3x^2\sin2x+2x^3\cos2x\)
\(=3x^2\sin2x+2x^3\cos2x\)
(6)\(y=\sin^{-1}2x\)
\(\displaystyle y'=\frac{(2x)'}{\sqrt{1-(2x)^2}}\)
\(\displaystyle =\frac{2}{\sqrt{1-4x^2}}\)
\(\displaystyle =\frac{2}{\sqrt{1-4x^2}}\)
(7)\(y=\sqrt{e^x+1}\)
\(\displaystyle y'=\frac{(e^x+1)'}{2\sqrt{e^x+1}}\)
\(\displaystyle =\frac{e^x}{2\sqrt{e^x+1}}\)
\(\displaystyle =\frac{e^x}{2\sqrt{e^x+1}}\)
(8)\(y=\{\sin(x+1)\}^3\)
\(y'=3\{\sin(x+1)\}^2\{\sin(x+1)\}'\)
\(=3\{\sin(x+1)\}^2\cos(x+1)(x+1)'\)
\(=3\{\sin(x+1)\}^2\cos(x+1)\)
\(=3\{\sin(x+1)\}^2\cos(x+1)(x+1)'\)
\(=3\{\sin(x+1)\}^2\cos(x+1)\)
(9)\(y=x\sin^{-1}2x\)
\(\displaystyle y'=\sin^{-1}2x+x\frac{1}{\sqrt{1-(2x)^2}}(2x)'\)
\(\displaystyle =\sin^{-1}2x+\frac{2x}{\sqrt{1-4x^2}}\)
\(\displaystyle =\sin^{-1}2x+\frac{2x}{\sqrt{1-4x^2}}\)
次の学習に進もう！