与えられた20個の関数を微分せよという問題です。

はい、承知いたしました。画像の数学の問題を解いていきます。

1. 問題の内容

与えられた20個の関数を微分せよという問題です。

2. 解き方の手順

(1) $y = 2x + 4$

$y' = 2$

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

$y' = 6x - 5$

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

$y' = x + 1$

(4) $y = 2x^3 + 4x^2 - 3x + 5$

$y' = 6x^2 + 8x - 3$

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

$y' = 5x^4 + 3x^2 + 1$

(6) $y = -4x^6 - 2x^4 + 3x^2 - 1$

$y' = -24x^5 - 8x^3 + 6x$

(7) $y = (3x+1)(x^2-4)$

$y = 3x^3 + x^2 - 12x - 4$

$y' = 9x^2 + 2x - 12$

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

$y = 2x^4 + 2x^3 + 2x^2 - 3x^2 - 3x - 3$

$y = 2x^4 + 2x^3 - x^2 - 3x - 3$

$y' = 8x^3 + 6x^2 - 2x - 3$

(9) $y = (x^3 - 2x)(3x^4 + 1)$

$y = 3x^7 + x^3 - 6x^5 - 2x$

$y' = 21x^6 + 3x^2 - 30x^4 - 2$

$y' = 21x^6 - 30x^4 + 3x^2 - 2$

(10) $y = \frac{3}{2x+1}$

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

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

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

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

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

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

$y' = -8x^{-3} = -\frac{8}{x^3}$

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

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

(15) $y = 2\cos x + 3\sin x + 4$

$y' = -2\sin x + 3\cos x$

(16) $y = 3e^x + 5\log x$

$y' = 3e^x + \frac{5}{x}$

(17) $y = \sin x \cdot \cos x = \frac{1}{2} \sin 2x$

$y' = \cos 2x$

または

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

(18) $y = x \cdot \log x$

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

(19) $y = x^2 \cdot e^x$

$y' = 2x \cdot e^x + x^2 \cdot e^x = (x^2 + 2x) e^x$

(20) $y = \frac{\sin x}{\cos x} = \tan x$

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

3. 最終的な答え

(1) $y' = 2$

(2) $y' = 6x - 5$

(3) $y' = x + 1$

(4) $y' = 6x^2 + 8x - 3$

(5) $y' = 5x^4 + 3x^2 + 1$

(6) $y' = -24x^5 - 8x^3 + 6x$

(7) $y' = 9x^2 + 2x - 12$

(8) $y' = 8x^3 + 6x^2 - 2x - 3$

(9) $y' = 21x^6 - 30x^4 + 3x^2 - 2$

(10) $y' = \frac{-6}{(2x+1)^2}$

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

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

(13) $y' = -\frac{8}{x^3}$

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

(15) $y' = -2\sin x + 3\cos x$

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

(17) $y' = \cos 2x$

(18) $y' = \log x + 1$

(19) $y' = (x^2 + 2x) e^x$

(20) $y' = \sec^2 x$