導関数の計算(calculation of derivatives)

確認問題

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) $\displaystyle{y = x^{\frac{1}{n}}, x > 0}$ (b) $\displaystyle{y = \sqrt{x}, x > 0}$
2.: 次の関数の導関数を求めよう．
(a) $\displaystyle{y = (x^{2} + 1)^{2004}}$ (b) $\displaystyle{y = (x^{2} + \frac{1}{x^{2}})^{3}}$ (c) $\displaystyle{y = [(2x+1)^{2} + (x+1)^{2}]^{3}}$
3.: $\frac{dy}{dx}$ を求めよう．
(a) $\displaystyle{x = t + 1, y = t^{2}-1}$ (b) $\displaystyle{x^{2} + y^{2} = 1}$
4.: 次の関数の導関数を求めよう．
(a) $\displaystyle{x^{2}\log{x}}$ (b) $\displaystyle{x^{3}\sin{2x}}$ (c) $\displaystyle{\sin^{-1}{(2x)}}$ (d) $\displaystyle{\sqrt{e^{x} + 1}}$ (e) $\displaystyle{(\sin(x+1))^{3}}$ (f) $\displaystyle{x\sin^{-1}(2x)}$

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) $\displaystyle{\log{\vert x + \sqrt{x^{2} + A}\vert}}$ (l) $\displaystyle{y = \sin{(x^{2} + 1)}}$ (m) $\displaystyle{y = \cos{(\sqrt{x + 1})}}$ (n) $\displaystyle{y = e^{\sin{x}}}$