次の関数を微分する問題です。 (1) $y = x^4$ (2) $y = x^6$ (3) $y = x^{100}$ (4) $y = \sqrt{x}$ ($x > 0$) (5) $y = \sqrt[4]{x^3}$ ($x > 0$) (6) $y = \frac{1}{x}$ (7) $y = \frac{1}{x^2}$ (8) $y = \frac{1}{x^4}$ (9) $y = \frac{1}{\sqrt[3]{x}}$ (10) $y = \frac{1}{x\sqrt{x}}$ ($x > 0$)

1. 問題の内容

次の関数を微分する問題です。

(1) $y = x^4$

(2) $y = x^6$

(3) $y = x^{100}$

(4) $y = \sqrt{x}$ ($x > 0$)

(5) $y = \sqrt[4]{x^3}$ ($x > 0$)

(6) $y = \frac{1}{x}$

(7) $y = \frac{1}{x^2}$

(8) $y = \frac{1}{x^4}$

(9) $y = \frac{1}{\sqrt[3]{x}}$

(10) $y = \frac{1}{x\sqrt{x}}$ ($x > 0$)

2. 解き方の手順

微分公式：$\frac{d}{dx}(x^n) = nx^{n-1}$、$\frac{d}{dx}(cf(x)) = c\frac{d}{dx}f(x)$（$c$は定数）を利用します。

(1) $y = x^4$

$\frac{dy}{dx} = 4x^{4-1} = 4x^3$

(2) $y = x^6$

$\frac{dy}{dx} = 6x^{6-1} = 6x^5$

(3) $y = x^{100}$

$\frac{dy}{dx} = 100x^{100-1} = 100x^{99}$

(4) $y = \sqrt{x} = x^{\frac{1}{2}}$

$\frac{dy}{dx} = \frac{1}{2}x^{\frac{1}{2}-1} = \frac{1}{2}x^{-\frac{1}{2}} = \frac{1}{2\sqrt{x}}$

(5) $y = \sqrt[4]{x^3} = x^{\frac{3}{4}}$

$\frac{dy}{dx} = \frac{3}{4}x^{\frac{3}{4}-1} = \frac{3}{4}x^{-\frac{1}{4}} = \frac{3}{4\sqrt[4]{x}}$

(6) $y = \frac{1}{x} = x^{-1}$

$\frac{dy}{dx} = -1x^{-1-1} = -x^{-2} = -\frac{1}{x^2}$

(7) $y = \frac{1}{x^2} = x^{-2}$

$\frac{dy}{dx} = -2x^{-2-1} = -2x^{-3} = -\frac{2}{x^3}$

(8) $y = \frac{1}{x^4} = x^{-4}$

$\frac{dy}{dx} = -4x^{-4-1} = -4x^{-5} = -\frac{4}{x^5}$

(9) $y = \frac{1}{\sqrt[3]{x}} = \frac{1}{x^{\frac{1}{3}}} = x^{-\frac{1}{3}}$

$\frac{dy}{dx} = -\frac{1}{3}x^{-\frac{1}{3}-1} = -\frac{1}{3}x^{-\frac{4}{3}} = -\frac{1}{3x^{\frac{4}{3}}} = -\frac{1}{3\sqrt[3]{x^4}} = -\frac{1}{3x\sqrt[3]{x}}$

(10) $y = \frac{1}{x\sqrt{x}} = \frac{1}{x \cdot x^{\frac{1}{2}}} = \frac{1}{x^{\frac{3}{2}}} = x^{-\frac{3}{2}}$

$\frac{dy}{dx} = -\frac{3}{2}x^{-\frac{3}{2}-1} = -\frac{3}{2}x^{-\frac{5}{2}} = -\frac{3}{2x^{\frac{5}{2}}} = -\frac{3}{2x^2\sqrt{x}}$

3. 最終的な答え

(1) $4x^3$

(2) $6x^5$

(3) $100x^{99}$

(4) $\frac{1}{2\sqrt{x}}$

(5) $\frac{3}{4\sqrt[4]{x}}$

(6) $-\frac{1}{x^2}$

(7) $-\frac{2}{x^3}$

(8) $-\frac{4}{x^5}$

(9) $-\frac{1}{3x\sqrt[3]{x}}$

(10) $-\frac{3}{2x^2\sqrt{x}}$