与えられた関数$y$について、その導関数$y'$を求めよ。

はい、承知いたしました。

1. 問題の内容

与えられた関数$y$について、その導関数$y'$を求めよ。

2. 解き方の手順

関数の導関数を求めるには、べき乗の微分公式を用います。

べき乗の微分公式は、$y = x^n$のとき、$y' = nx^{n-1}$となります。

係数がある場合は、係数はそのままにして、べき乗の部分を微分します。

各問題について具体的に見ていきましょう。

(5) $y = x^6$

$y' = 6x^{6-1} = 6x^5$

(6) $y = x^7$

$y' = 7x^{7-1} = 7x^6$

(7) $y = x^8$

$y' = 8x^{8-1} = 8x^7$

(8) $y = x^9$

$y' = 9x^{9-1} = 9x^8$

(9) $y = x^{10}$

$y' = 10x^{10-1} = 10x^9$

(10) $y = x^{17}$

$y' = 17x^{17-1} = 17x^{16}$

(11) $y = x^{20}$

$y' = 20x^{20-1} = 20x^{19}$

(12) $y = x^{25}$

$y' = 25x^{25-1} = 25x^{24}$

(13) $y = x^{29}$

$y' = 29x^{29-1} = 29x^{28}$

(14) $y = x^{100}$

$y' = 100x^{100-1} = 100x^{99}$

(15) $y = x^{123}$

$y' = 123x^{123-1} = 123x^{122}$

(16) $y = x^{150}$

$y' = 150x^{150-1} = 150x^{149}$

(17) $y = 3x^2$

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

(18) $y = 5x^3$

$y' = 5 * (3x^{3-1}) = 15x^2$

(19) $y = -2x^4$

$y' = -2 * (4x^{4-1}) = -8x^3$

(20) $y = -x^5$

$y' = -(5x^{5-1}) = -5x^4$

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

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

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

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

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

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

(24) $y = \frac{2}{3}x^6$

$y' = \frac{2}{3} * (6x^{6-1}) = 4x^5$

(25) $y = -\frac{1}{3}x^9$

$y' = -\frac{1}{3} * (9x^{9-1}) = -3x^8$

(26) $y = -\frac{1}{8}x^4$

$y' = -\frac{1}{8} * (4x^{4-1}) = -\frac{1}{2}x^3$

3. 最終的な答え

(5) $y' = 6x^5$

(6) $y' = 7x^6$

(7) $y' = 8x^7$

(8) $y' = 9x^8$

(9) $y' = 10x^9$

(10) $y' = 17x^{16}$

(11) $y' = 20x^{19}$

(12) $y' = 25x^{24}$

(13) $y' = 29x^{28}$

(14) $y' = 100x^{99}$

(15) $y' = 123x^{122}$

(16) $y' = 150x^{149}$

(17) $y' = 6x$

(18) $y' = 15x^2$

(19) $y' = -8x^3$

(20) $y' = -5x^4$

(21) $y' = x$

(22) $y' = x^2$

(23) $y' = 2x^3$

(24) $y' = 4x^5$

(25) $y' = -3x^8$

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