2.2.1 解答

2.2.1

(a) $\displaystyle{\left(\frac{3x - 1}{x^2 + 1}\right)^{\prime} = \frac{3(x^2 + 1) - (3x-1)(2x)}{(x^2 + 1)^2} = \frac{-3x^2 + 2x + 3}{(x^2 + 1)^2}}$

(b) $\displaystyle{(\sec{x})^{\prime} = \left(\frac{1}{\cos{x}}\right)^{\prime} = \sec{x}\tan{x}}$

(d) $\displaystyle{(\cot{x})^{\prime} = \left(\frac{\cos{x}}{\sin{x}}\right)^{\prime} = -\frac{1}{\sin^{2}{x}}}$

(a) $\displaystyle{(\sin{(x^{2} + 1)})^{\prime} = 2x\cos{(x^2 + 1)}}$

$\displaystyle{(\cos{\sqrt{x+ 1}})^{\prime} = -\sin{\sqrt{x+1}}(\sqrt{x+1})^{\prime} = -\frac{\sin{(\sqrt{x+1})}}{2\sqrt{x+1}}}$

演習問題

1.: 次の関数の導関数を逆関数の微分法を用いて求めよう．
(a): $\displaystyle{y = \cos^{-1}{x}}$
(b): $\displaystyle{y = \tan^{-1}{x}}$
2.: 次の関数の導関数を対数微分法を用いて求めよう．
(a): $\displaystyle{y = x^{2}\sqrt{\frac{x^{3} + 2x + 1}{x^{2} - 3x + 1}}}$
(b): $\displaystyle{y = x^{x}}$
(c): $\displaystyle{y = \sin{x}^{x}}$
(d): $\displaystyle{y = x^{1/x}}$
3.: $\frac{dy}{dx}$ を求めよう．
(a): $\displaystyle{x = a\cos{t}, y = a\sin{t}, a > 0}$
(b): $\displaystyle{x = \sqrt{t} - \frac{1}{t}, y = t + \frac{1}{\sqrt{t}}}$
4.: 次の関数の導関数を求めよう．
(a): $\displaystyle{x^{2}(1 + \sqrt{x})}$
(b): $\displaystyle{x^{3}\tan{2x}}$
(c): $\displaystyle{x\sin^{-1}{x}}$
(d): $\displaystyle{\frac{x}{x^{2}+1}}$
(e): $\displaystyle{x\sin{x}}$
(f): $\displaystyle{x\sin^{-1}x + \sqrt{1-x^{2}}}$
(g): $\displaystyle{\tan^{-1}(x^{2} + 1)}$
(h): $\displaystyle{\cos{(\sqrt{2x+1})}}$
(i): $\displaystyle{\frac{\sin{x} - x\cos{x}}{x\sin{x} + \cos{x}}}$
(j): $\displaystyle{e^{2x}\cos{x}}$
(k): $\displaystyle{\log{\vert x + \sqrt{x^{2} + A}\vert}}$