【微分積分】3-4-3 n階導関数の計算｜要点まとめ

\(\displaystyle f^{(n)}(x)=2^x(\log2)^n\)

\(\displaystyle f(x)=\frac{1}{2}\left(\frac{1}{x-3}-\frac{1}{x-1}\right)\)
\(\displaystyle f^{(n)}(x)=\frac{1}{2}\left\{\frac{(-1)^nn!}{(x-3)^{n+1}}-\frac{(-1)^nn!}{(x-1)^{n+1}}\right\}\)
\(\displaystyle =\frac{(-1)^nn!}{2}\left\{\frac{1}{(x-3)^{n+1}}-\frac{1}{(x-1)^{n+1}}\right\}\)

\(\displaystyle f(x)=\frac{1}{2}\sin2x\)
\(\displaystyle f^{(n)}(x)=\frac{1}{2}・2^n\sin\left(2x+\frac{n\pi}{2}\right)\)
\(\displaystyle =2^{n-1}\sin\left(2x+\frac{n\pi}{2}\right)\)

\(\displaystyle f(x)=\frac{3\sin x-\sin3x}{4}\)
\(f^{(n)}(x)\)
\(\displaystyle =\frac{3}{4}\sin\left(x+\frac{n\pi}{2}\right)-\frac{3^n}{4}\sin\left(3x+\frac{n\pi}{2}\right)\)

\(f'(x)=e^x\sin x+e^x\cos x\)
\(=e^x(\sin x+\cos x)\)
\(\displaystyle =\sqrt{2}e^x\sin\left(x+\frac{\pi}{4}\right)\)
\(\displaystyle f^{(n)}(x)=(\sqrt{2})^ne^x\sin\left(x+\frac{n\pi}{4}\right)\)