以下の10個の関数を微分する問題です。 (1) $y = x - 3$ (2) $y = \log_3 x$ (3) $y = \sin x$ (4) $y = \frac{\sin x}{\cos x}$ (5) $y = e^{3x}$ (6) $y = \log |\sin x|$ (7) $y = (3x+5)^4$ (8) $y = \sqrt[5]{x^4}$ (9) $y = 2^x$ (10) $y = \sin^2 3x$

1. 問題の内容

以下の10個の関数を微分する問題です。

(1) $y = x - 3$

(2) $y = \log_3 x$

(3) $y = \sin x$

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

(5) $y = e^{3x}$

(6) $y = \log |\sin x|$

(7) $y = (3x+5)^4$

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

(9) $y = 2^x$

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

2. 解き方の手順

各関数の微分を計算します。

(1) $y = x - 3$ の微分：

$y' = \frac{d}{dx}(x - 3) = 1 - 0 = 1$

(2) $y = \log_3 x$ の微分：

$y' = \frac{d}{dx}(\log_3 x) = \frac{1}{x \ln 3}$

(3) $y = \sin x$ の微分：

$y' = \frac{d}{dx}(\sin x) = \cos x$

(4) $y = \frac{\sin x}{\cos x} = \tan x$ の微分：

$y' = \frac{d}{dx}(\tan x) = \frac{1}{\cos^2 x} = \sec^2 x$

(5) $y = e^{3x}$ の微分：

$y' = \frac{d}{dx}(e^{3x}) = 3e^{3x}$

(6) $y = \log |\sin x|$ の微分：

$y' = \frac{d}{dx}(\log |\sin x|) = \frac{1}{\sin x} \cdot \cos x = \cot x$

(7) $y = (3x+5)^4$ の微分：

$y' = \frac{d}{dx}((3x+5)^4) = 4(3x+5)^3 \cdot 3 = 12(3x+5)^3$

(8) $y = \sqrt[5]{x^4} = x^{\frac{4}{5}}$ の微分：

$y' = \frac{d}{dx}(x^{\frac{4}{5}}) = \frac{4}{5} x^{\frac{4}{5} - 1} = \frac{4}{5} x^{-\frac{1}{5}} = \frac{4}{5 \sqrt[5]{x}}$

(9) $y = 2^x$ の微分：

$y' = \frac{d}{dx}(2^x) = 2^x \ln 2$

(10) $y = \sin^2 3x$ の微分：

$y' = \frac{d}{dx}(\sin^2 3x) = 2 \sin 3x \cdot \cos 3x \cdot 3 = 6 \sin 3x \cos 3x = 3 \sin 6x$

3. 最終的な答え

(1) $y' = 1$

(2) $y' = \frac{1}{x \ln 3}$

(3) $y' = \cos x$

(4) $y' = \sec^2 x$

(5) $y' = 3e^{3x}$

(6) $y' = \cot x$

(7) $y' = 12(3x+5)^3$

(8) $y' = \frac{4}{5 \sqrt[5]{x}}$

(9) $y' = 2^x \ln 2$

(10) $y' = 3 \sin 6x$