与えられた10個の関数を微分する問題です。 (1) $y = \frac{\cos x}{\sin x}$ (2) $y = \cos(3x + 2)$ (3) $y = e^{2x + 3}$ (4) $y = xe^{2x}$ (5) $y = \sqrt[3]{e^{2x}}$ (6) $y = x^2 \sin(3x)$ (7) $y = \frac{\log x}{x^2}$ (8) $y = 2^{3x + 1}$ (9) $y = \log_5(2x - 3)$ (10) $y = \log|4 - 7x|$

はい、承知いたしました。与えられた関数の微分を求めます。

1. 問題の内容

与えられた10個の関数を微分する問題です。

(1) $y = \frac{\cos x}{\sin x}$

(2) $y = \cos(3x + 2)$

(3) $y = e^{2x + 3}$

(4) $y = xe^{2x}$

(5) $y = \sqrt[3]{e^{2x}}$

(6) $y = x^2 \sin(3x)$

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

(8) $y = 2^{3x + 1}$

(9) $y = \log_5(2x - 3)$

(10) $y = \log|4 - 7x|$

2. 解き方の手順

(1) $y = \frac{\cos x}{\sin x} = \cot x$

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

(2) $y = \cos(3x + 2)$

$\frac{dy}{dx} = -\sin(3x + 2) \cdot 3 = -3\sin(3x + 2)$

(3) $y = e^{2x + 3}$

$\frac{dy}{dx} = e^{2x + 3} \cdot 2 = 2e^{2x + 3}$

(4) $y = xe^{2x}$

$\frac{dy}{dx} = e^{2x} + x \cdot e^{2x} \cdot 2 = e^{2x}(1 + 2x)$

(5) $y = \sqrt[3]{e^{2x}} = e^{\frac{2x}{3}}$

$\frac{dy}{dx} = e^{\frac{2x}{3}} \cdot \frac{2}{3} = \frac{2}{3}e^{\frac{2x}{3}}$

(6) $y = x^2 \sin(3x)$

$\frac{dy}{dx} = 2x \sin(3x) + x^2 \cos(3x) \cdot 3 = 2x \sin(3x) + 3x^2 \cos(3x)$

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

$\frac{dy}{dx} = \frac{\frac{1}{x} \cdot x^2 - \log x \cdot 2x}{x^4} = \frac{x - 2x \log x}{x^4} = \frac{1 - 2 \log x}{x^3}$

(8) $y = 2^{3x + 1}$

$\frac{dy}{dx} = 2^{3x + 1} \cdot \log 2 \cdot 3 = 3 \cdot 2^{3x + 1} \log 2$

(9) $y = \log_5(2x - 3)$

$\frac{dy}{dx} = \frac{1}{(2x - 3) \log 5} \cdot 2 = \frac{2}{(2x - 3) \log 5}$

(10) $y = \log|4 - 7x|$

$\frac{dy}{dx} = \frac{1}{4 - 7x} \cdot (-7) = \frac{-7}{4 - 7x} = \frac{7}{7x - 4}$

3. 最終的な答え

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

(2) $\frac{dy}{dx} = -3\sin(3x + 2)$

(3) $\frac{dy}{dx} = 2e^{2x + 3}$

(4) $\frac{dy}{dx} = e^{2x}(1 + 2x)$

(5) $\frac{dy}{dx} = \frac{2}{3}e^{\frac{2x}{3}}$

(6) $\frac{dy}{dx} = 2x \sin(3x) + 3x^2 \cos(3x)$

(7) $\frac{dy}{dx} = \frac{1 - 2 \log x}{x^3}$

(8) $\frac{dy}{dx} = 3 \cdot 2^{3x + 1} \log 2$

(9) $\frac{dy}{dx} = \frac{2}{(2x - 3) \log 5}$

(10) $\frac{dy}{dx} = \frac{7}{7x - 4}$