Задание 3. Найдите производную

Для решения этих задач воспользуемся правилами дифференцирования: - $(x^n)' = nx^{n-1}$ - $(cf)' = c \cdot f'$ - $(f \pm g)' = f' \pm g'$ - $(fg)' = f'g + fg'$ - $(\frac{f}{g})' = \frac{f'g - fg'}{g^2}$ 1. $f'(x) = 21x^2 - 4x + 1$ 2. $f'(x) = 2x^7 + \frac{3}{2\sqrt{x}} - 5$ 3. $f'(x) = -4x^{-3} - x^2 = -\frac{4}{x^3} - x^2$ 4. $f'(x) = \frac{2}{5}x^{-3/5} + \frac{1}{\cos^2 x} = \frac{2}{5\sqrt[5]{x^3}} + \frac{1}{\cos^2 x}$ 5. $f'(x) = -15x^{-4} + 9x^8 = -\frac{15}{x^4} + 9x^8$ 6. $f(x) = x^3 - 5x^2 + 9x + 3x^2 - 15x + 27 = x^3 - 2x^2 - 6x + 27$. $f'(x) = 3x^2 - 4x - 6$ 7. $f'(x) = (8x)\sqrt{x} + (4x^2 - 1)\frac{1}{2\sqrt{x}} = 8x\sqrt{x} + \frac{4x^2 - 1}{2\sqrt{x}} = \frac{16x^2 + 4x^2 - 1}{2\sqrt{x}} = \frac{20x^2 - 1}{2\sqrt{x}}$ 8. $f'(x) = \frac{1}{\cos^2 x}(3 - 2x) + \tg x(-2) = \frac{3-2x}{\cos^2 x} - 2\tg x$ 9. $f'(x) = 2x \sin x + x^2 \cos x$ 10. $f'(x) = \sqrt{3} \cos x + \sin x + 9x^2$ 11. $f'(x) = \frac{4(8-2x) - (4x-7)(-2)}{(8-2x)^2} = \frac{32 - 8x + 8x - 14}{(8-2x)^2} = \frac{18}{(8-2x)^2}$ 12. $f'(x) = \frac{\sin x(1+\cos x) - (1-\cos x)(-\sin x)}{(1+\cos x)^2} = \frac{\sin x + \sin x \cos x + \sin x - \sin x \cos x}{(1+\cos x)^2} = \frac{2\sin x}{(1+\cos x)^2}$ 13. $f(x) = 2x^{1/2} - 5x^{-1/2}$. $f'(x) = x^{-1/2} + \frac{5}{2}x^{-3/2} = \frac{1}{\sqrt{x}} + \frac{5}{2x\sqrt{x}} = \frac{2x+5}{2x\sqrt{x}}$ 14. $f'(x) = \frac{-2x(x^6-5) - (-x^2)(6x^5)}{(x^6-5)^2} = \frac{-2x^7 + 10x + 6x^7}{(x^6-5)^2} = \frac{4x^7 + 10x}{(x^6-5)^2}$ 15. $f'(x) = \frac{\cos x \cdot 7x - \sin x \cdot 7}{(7x)^2} = \frac{7x \cos x - 7 \sin x}{49x^2} = \frac{x \cos x - \sin x}{7x^2}$