【微分積分】3-4-1 高階導関数｜問題集
1.次の関数の\(2\)階導関数を答えなさい。
(1)\(f(x)=4+3x-x^2\)
\(f'(x)=3-2x\)
\(f''(x)=-2\)
\(f''(x)=-2\)
(2)\(f(x)=x^3-6x\)
\(f'(x)=3x^2-6\)
\(f''(x)=6x\)
\(f''(x)=6x\)
(3)\(\displaystyle f(x)=\frac{18}{x+2}\)
\(\displaystyle f'(x)=-\frac{18}{(x+2)^2}\)
\(\displaystyle f''(x)=\frac{36}{(x+2)^3}\)
\(\displaystyle f''(x)=\frac{36}{(x+2)^3}\)
(4)\(f(x)=\sqrt{x^2+1}\)
\(\displaystyle f'(x)=\frac{x}{\sqrt{x^2+1}}\)
\(\displaystyle f''(x)=\frac{1}{(\sqrt{x^2+1})^3}\)
\(\displaystyle f''(x)=\frac{1}{(\sqrt{x^2+1})^3}\)
(5)\(f(x)=x\log x\)
\(f'(x)=\log x+1\)
\(\displaystyle f''(x)=\frac{1}{x}\)
\(\displaystyle f''(x)=\frac{1}{x}\)
(6)\(f(x)=e^x\sin x\)
\(f'(x)=e^x(\sin x+\cos x)\)
\(f''(x)=2e^x\cos x\)
\(f''(x)=2e^x\cos x\)
2.次の関数の\(n\)階導関数を答えなさい。
(1)\(f(x)=\log(1+x)\)
\(\displaystyle f^{(n)}(x)=\frac{(-1)^{n-1}(n-1)!}{(1+x)^n}\)
3.関数\(f(x)\)は\(2\)回微分可能で、さらに逆関数\(g(x)\)をもつ。\(f(1)=2\)、\(f'(1)=2\)、\(f''(1)=3\)のとき、\(g''(2)\)の値を求めなさい。
逆関数の定義より
\(f(g(x))=x\)
\(\displaystyle g'(x)=\frac{1}{f'(g(x))}\)
\(\displaystyle g''(x)=-\frac{f''(g(x))}{(f'(g(x)))^3}\)
\(f(1)=2\)なので、\(g(2)=1\)
\(\displaystyle g''(2)=-\frac{f''(1)}{(f'(1))^3}=-\frac{3}{2^3}=-\frac{3}{8}\)
\(f(g(x))=x\)
\(\displaystyle g'(x)=\frac{1}{f'(g(x))}\)
\(\displaystyle g''(x)=-\frac{f''(g(x))}{(f'(g(x)))^3}\)
\(f(1)=2\)なので、\(g(2)=1\)
\(\displaystyle g''(2)=-\frac{f''(1)}{(f'(1))^3}=-\frac{3}{2^3}=-\frac{3}{8}\)
次の学習に進もう！