次の関数を微分する問題です。 (a) $y = 5^x$ (b) $y = 3^{-x}$ (c) $y = 4^{\frac{1}{x}}$ (d) $y = 3^x \log x$ (e) $y = e^{3x^2 - x}$ (f) $y = (x^2 - 3x)e^x$ (g) $y = e^{-x} \sin 2x$ (h) $y = e^{2x} \sin^{-1} x$ (i) $y = \frac{e^{2x}}{x+1}$

1. 問題の内容

次の関数を微分する問題です。

(a) $y = 5^x$

(b) $y = 3^{-x}$

(c) $y = 4^{\frac{1}{x}}$

(d) $y = 3^x \log x$

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

(f) $y = (x^2 - 3x)e^x$

(g) $y = e^{-x} \sin 2x$

(h) $y = e^{2x} \sin^{-1} x$

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

2. 解き方の手順

(a) $y = 5^x$

$y' = 5^x \log 5$

(b) $y = 3^{-x}$

$y' = 3^{-x} \log 3 \cdot (-1) = -3^{-x} \log 3$

(c) $y = 4^{\frac{1}{x}}$

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

(d) $y = 3^x \log x$

積の微分法を用いる。$(uv)' = u'v + uv'$

$u = 3^x$, $u' = 3^x \log 3$

$v = \log x$, $v' = \frac{1}{x}$

$y' = 3^x \log 3 \cdot \log x + 3^x \cdot \frac{1}{x} = 3^x \log 3 \log x + \frac{3^x}{x}$

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

合成関数の微分法を用いる。

$y' = e^{3x^2 - x} \cdot (6x - 1) = (6x - 1)e^{3x^2 - x}$

(f) $y = (x^2 - 3x)e^x$

積の微分法を用いる。

$u = x^2 - 3x$, $u' = 2x - 3$

$v = e^x$, $v' = e^x$

$y' = (2x - 3)e^x + (x^2 - 3x)e^x = (x^2 - x - 3)e^x$

(g) $y = e^{-x} \sin 2x$

積の微分法を用いる。

$u = e^{-x}$, $u' = -e^{-x}$

$v = \sin 2x$, $v' = 2\cos 2x$

$y' = -e^{-x} \sin 2x + e^{-x} \cdot 2\cos 2x = -e^{-x} \sin 2x + 2e^{-x} \cos 2x = e^{-x}(2\cos 2x - \sin 2x)$

(h) $y = e^{2x} \sin^{-1} x$

積の微分法を用いる。

$u = e^{2x}$, $u' = 2e^{2x}$

$v = \sin^{-1} x$, $v' = \frac{1}{\sqrt{1-x^2}}$

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

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

商の微分法を用いる。$(\frac{u}{v})' = \frac{u'v - uv'}{v^2}$

$u = e^{2x}$, $u' = 2e^{2x}$

$v = x+1$, $v' = 1$

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

3. 最終的な答え

(a) $y' = 5^x \log 5$

(b) $y' = -3^{-x} \log 3$

(c) $y' = -\frac{4^{\frac{1}{x}} \log 4}{x^2}$

(d) $y' = 3^x \log 3 \log x + \frac{3^x}{x}$

(e) $y' = (6x - 1)e^{3x^2 - x}$

(f) $y' = (x^2 - x - 3)e^x$

(g) $y' = e^{-x}(2\cos 2x - \sin 2x)$

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

(i) $y' = \frac{e^{2x}(2x+1)}{(x+1)^2}$