以下の関数を微分せよ。 (1) $(x^2 + x + 1)^3$ (2) $(3x + 7)^5$ (3) $(ax + b)^n$ (4) $\frac{1}{x^2 + x + 1}$ (5) $\frac{1}{(x^2 + x + 1)^3}$ (6) $(x + \frac{1}{x})^4$ (7) $\sqrt{x^2 + 3x + 7}$ (8) $\sqrt{x + \sqrt{x^2 + 1}}$ (9) $(\frac{x^2}{2x - 3})^4$

はい、承知いたしました。画像の問題を解きます。

1. 問題の内容

以下の関数を微分せよ。

(1) $(x^2 + x + 1)^3$

(2) $(3x + 7)^5$

(3) $(ax + b)^n$

(4) $\frac{1}{x^2 + x + 1}$

(5) $\frac{1}{(x^2 + x + 1)^3}$

(6) $(x + \frac{1}{x})^4$

(7) $\sqrt{x^2 + 3x + 7}$

(8) $\sqrt{x + \sqrt{x^2 + 1}}$

(9) $(\frac{x^2}{2x - 3})^4$

2. 解き方の手順

各関数に対して、微分公式と合成関数の微分法（連鎖律）を適用します。

(1) $y = (x^2 + x + 1)^3$

$y' = 3(x^2 + x + 1)^2 \cdot (2x + 1) = (6x + 3)(x^2 + x + 1)^2$

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

$y' = 5(3x + 7)^4 \cdot 3 = 15(3x + 7)^4$

(3) $y = (ax + b)^n$

$y' = n(ax + b)^{n-1} \cdot a = an(ax + b)^{n-1}$

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

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

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

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

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

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

(7) $y = \sqrt{x^2 + 3x + 7} = (x^2 + 3x + 7)^{\frac{1}{2}}$

$y' = \frac{1}{2}(x^2 + 3x + 7)^{-\frac{1}{2}} \cdot (2x + 3) = \frac{2x + 3}{2\sqrt{x^2 + 3x + 7}}$

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

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

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

$y' = 4(\frac{x^2}{2x - 3})^3 \cdot \frac{2x(2x - 3) - x^2 \cdot 2}{(2x - 3)^2} = 4(\frac{x^2}{2x - 3})^3 \cdot \frac{4x^2 - 6x - 2x^2}{(2x - 3)^2} = 4(\frac{x^2}{2x - 3})^3 \cdot \frac{2x^2 - 6x}{(2x - 3)^2} = \frac{8x^7(x - 3)}{(2x - 3)^5}$

3. 最終的な答え

(1) $(6x + 3)(x^2 + x + 1)^2$

(2) $15(3x + 7)^4$

(3) $an(ax + b)^{n-1}$

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

(5) $-\frac{3(2x + 1)}{(x^2 + x + 1)^4}$

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

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

(8) $\frac{1}{2\sqrt{x + \sqrt{x^2 + 1}}}(1 + \frac{x}{\sqrt{x^2 + 1}})$

(9) $\frac{8x^7(x - 3)}{(2x - 3)^5}$