与えられた関数 $y$ を微分せよ。 (1) $y = e^{2x}e^{4x}$ (2) $y = \frac{1}{e^{3x}}$ (3) $y = \frac{e^{x}}{e^{5x}}$ (4) $y = \sqrt{e^{8x}}$ (5) $y = \frac{1}{\sqrt[3]{e^{6x}}}$ (6) $y = e^{3x}\sqrt{e^{5x}}$

1. 問題の内容

与えられた関数 $y$ を微分せよ。

(1) $y = e^{2x}e^{4x}$

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

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

(4) $y = \sqrt{e^{8x}}$

(5) $y = \frac{1}{\sqrt[3]{e^{6x}}}$

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

2. 解き方の手順

各関数に対して以下の手順で微分を行う。

(1) $y = e^{2x}e^{4x} = e^{2x+4x} = e^{6x}$

$\frac{dy}{dx} = 6e^{6x}$

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

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

(3) $y = \frac{e^{x}}{e^{5x}} = e^{x-5x} = e^{-4x}$

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

(4) $y = \sqrt{e^{8x}} = (e^{8x})^{\frac{1}{2}} = e^{4x}$

$\frac{dy}{dx} = 4e^{4x}$

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

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

(6) $y = e^{3x}\sqrt{e^{5x}} = e^{3x}(e^{5x})^{\frac{1}{2}} = e^{3x}e^{\frac{5}{2}x} = e^{(3+\frac{5}{2})x} = e^{\frac{11}{2}x}$

$\frac{dy}{dx} = \frac{11}{2}e^{\frac{11}{2}x}$

3. 最終的な答え

(1) $\frac{dy}{dx} = 6e^{6x}$

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

(3) $\frac{dy}{dx} = -4e^{-4x}$

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

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

(6) $\frac{dy}{dx} = \frac{11}{2}e^{\frac{11}{2}x}$