【微分積分】3-4-2 ライプニッツの公式｜要点まとめ

\(\displaystyle f^{(n)}(x)=\sum_{k=0}^{n}{}_{n}\mathrm{C}_k(x)^{(k)}(e^{2x})^{(n-k)}\)
\(=x(e^{2x})^{(n)}+{}_{n}\mathrm{C}_1(x)'(e^{2x})^{(n-1)}\)
\(=x・2^ne^{2x}+n・1・2^{n-1}e^{2x}\)
\(=2^{n-1}(2x+n)e^{2x}\)

\(\displaystyle f^{(n)}(x)=\sum_{k=0}^{n}{}_{n}\mathrm{C}_k(x^2)^{(k)}(e^x)^{(n-k)}\)
\(=x^2(e^x)^{(n)}+{}_{n}\mathrm{C}_1(x^2)'(e^{x})^{(n-1)}\)
\(\ \ \ \ \ +{}_{n}\mathrm{C}_2(x^2)''(e^{x})^{(n-2)}\)
\(\displaystyle =x^2・e^x+n・2x・e^x+\frac{n(n-1)}{2}・2e^x\)
\(=e^x\{x^2+2nx+n(n-1)\}\)

\(\displaystyle f^{(n)}(x)=\sum_{k=0}^{n}{}_{n}\mathrm{C}_k(x^2)^{(k)}(\sin x)^{(n-k)}\)
\(\displaystyle =x^2(\sin x)^{(n)}+{}_{n}\mathrm{C}_1(x^2)'(\sin x)^{(n-1)}\)
\(\ \ \ \ \ +{}_{n}\mathrm{C}_2(x^2)''(\sin x)^{(n-2)}\)
\(\displaystyle =x^2\sin\left(x+\frac{n\pi}{2}\right)\)
\(\displaystyle \ \ \ \ \ +n・2x\sin\left(x+\frac{n-1}{2}\pi\right)\)
\(\displaystyle \ \ \ \ \ +\frac{n(n-1)}{2}・2\sin\left(x+\frac{n-2}{2}\pi\right)\)
\(\displaystyle =\{x^2-n(n-1)\}\sin\left(x+\frac{n\pi}{2}\right)\)
\(\displaystyle \ \ \ \ \ -2nx\cos\left(x+\frac{n\pi}{2}\right)\)