第3章 3.4

(1)

$\cos 2x$

$f = \cos 2x$ とおくと

$f' = -2\sin 2x \qquad f'' = -4\cos 2x \qquad f''' = 8\sin 2x \qquad \text{これから} \qquad f^{(n)} = 2^{n}\cos\left(2x + \frac{\pi}{2}n\right)$

(2)

$\sin^{2}x$

$f = \sin^{2}x$ とおくと

$f' = 2\sin x \cdot \cos x = \sin 2x$

$f'' = 2\cos 2x \qquad f''' = -4\sin 2x \qquad \text{これから} \qquad f^{(n)} = 2^{n-1}\sin\left\{2x + \frac{\pi}{2}(n-1)\right\}$

(3)

$a^{x}$

$f = a^{x}$ とおくと

$f' = a^{x}\log a \qquad f'' = a^{x}\log a \cdot \log a = a^{x}(\log a)^{2} \qquad \text{これから} \qquad f^{(n)} = a^{x}(\log a)^{n}$

(4)

$x\log x$

$f = x\log x$ とおくと

$f' = \log x + x \cdot \frac{1}{x} = \log x + 1 \qquad f'' = \frac{1}{x} \qquad f''' = -\frac{1}{x^{2}}$

これから

$f' = \log x + 1 \qquad f^{(n)} = (-1)^{n}\frac{(n-2)!}{x^{n-1}}$