次の指数関数、対数関数を $x$ で微分せよ。 (1) $y = \frac{e^x}{3}$ (2) $y = -2e^x$ (3) $y = 5^x$ (4) $y = 0.25^x$ (5) $y = (\frac{3}{4})^x$ (6) $y = \log_e x^3$ (7) $y = \log_e (\frac{1}{x})$ (8) $y = \log_{10} x^2$ (9) $y = \frac{\log_{10} x}{2}$

1. 問題の内容

次の指数関数、対数関数を $x$ で微分せよ。

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

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

(3) $y = 5^x$

(4) $y = 0.25^x$

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

(6) $y = \log_e x^3$

(7) $y = \log_e (\frac{1}{x})$

(8) $y = \log_{10} x^2$

(9) $y = \frac{\log_{10} x}{2}$

2. 解き方の手順

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

$y' = \frac{1}{3}e^x$

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

$y' = -2e^x$

(3) $y = 5^x$

$y' = 5^x \log 5$

(4) $y = 0.25^x = (\frac{1}{4})^x$

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

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

$y' = (\frac{3}{4})^x \log (\frac{3}{4})$

(6) $y = \log_e x^3 = 3 \log_e x$

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

(7) $y = \log_e (\frac{1}{x}) = \log_e (x^{-1}) = -\log_e x$

$y' = -\frac{1}{x}$

(8) $y = \log_{10} x^2 = 2 \log_{10} x$

$y' = 2 \cdot \frac{1}{x \log 10} = \frac{2}{x \log 10}$

(9) $y = \frac{\log_{10} x}{2} = \frac{1}{2} \log_{10} x$

$y' = \frac{1}{2} \cdot \frac{1}{x \log 10} = \frac{1}{2x \log 10}$

3. 最終的な答え

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

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

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

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

(5) $y' = (\frac{3}{4})^x \log (\frac{3}{4})$

(6) $y' = \frac{3}{x}$

(7) $y' = -\frac{1}{x}$

(8) $y' = \frac{2}{x \log 10}$

(9) $y' = \frac{1}{2x \log 10}$