次の関数を微分せよ。 (1) $y = x^4$ (2) $y = \frac{1}{\sqrt{x^3}}$ (3) $y = x\sqrt[3]{x}$ (4) $y = 4x^{\frac{3}{2}} - 6x^{\frac{-1}{2}}$ (5) $y = 3x^{\frac{2}{3}} - \frac{1}{x^{\frac{1}{3}}\sqrt{x}}$

はい、承知いたしました。画像にある問題を解きます。

1. 問題の内容

次の関数を微分せよ。

(1) $y = x^4$

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

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

(4) $y = 4x^{\frac{3}{2}} - 6x^{\frac{-1}{2}}$

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

2. 解き方の手順

(1) $y = x^4$ の微分

$y' = 4x^{4-1} = 4x^3$

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

$y' = -\frac{3}{2}x^{-\frac{3}{2} - 1} = -\frac{3}{2}x^{-\frac{5}{2}} = -\frac{3}{2x^{\frac{5}{2}}} = -\frac{3}{2\sqrt{x^5}}$

(3) $y = x\sqrt[3]{x} = x \cdot x^{\frac{1}{3}} = x^{\frac{4}{3}}$ の微分

$y' = \frac{4}{3}x^{\frac{4}{3} - 1} = \frac{4}{3}x^{\frac{1}{3}} = \frac{4\sqrt[3]{x}}{3}$

(4) $y = 4x^{\frac{3}{2}} - 6x^{-\frac{1}{2}}$ の微分

$y' = 4 \cdot \frac{3}{2}x^{\frac{3}{2} - 1} - 6 \cdot (-\frac{1}{2})x^{-\frac{1}{2} - 1} = 6x^{\frac{1}{2}} + 3x^{-\frac{3}{2}} = 6\sqrt{x} + \frac{3}{x^{\frac{3}{2}}} = 6\sqrt{x} + \frac{3}{\sqrt{x^3}}$

(5) $y = 3x^{\frac{2}{3}} - \frac{1}{x^{\frac{1}{3}}\sqrt{x}} = 3x^{\frac{2}{3}} - \frac{1}{x^{\frac{1}{3}}x^{\frac{1}{2}}} = 3x^{\frac{2}{3}} - \frac{1}{x^{\frac{5}{6}}} = 3x^{\frac{2}{3}} - x^{-\frac{5}{6}}$ の微分

$y' = 3 \cdot \frac{2}{3}x^{\frac{2}{3} - 1} - (-\frac{5}{6})x^{-\frac{5}{6} - 1} = 2x^{-\frac{1}{3}} + \frac{5}{6}x^{-\frac{11}{6}} = \frac{2}{\sqrt[3]{x}} + \frac{5}{6x^{\frac{11}{6}}} = \frac{2}{\sqrt[3]{x}} + \frac{5}{6\sqrt[6]{x^{11}}}$

3. 最終的な答え

(1) $y' = 4x^3$

(2) $y' = -\frac{3}{2\sqrt{x^5}}$

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

(4) $y' = 6\sqrt{x} + \frac{3}{\sqrt{x^3}}$

(5) $y' = \frac{2}{\sqrt[3]{x}} + \frac{5}{6\sqrt[6]{x^{11}}}$