Самостоятельная работа №1 по теме «Производная» Вариант № 4 Найти производные следующих функций: 1. y = 8 x25 -3 2. y = 15 xs 3. y = 12x³- 2x + 14 4. y = -2-7x 1-3x 5. y = (3x5 + 7x² + 10x - 7)4 6. у = найти у (1) x6 7. y = √2x3-4 8. y = √2x (3x + 1) 9. y = 74x+1 10. y = ln(2x³ + 6) 11. y = ctg³ x + cos 2x4 12. y = 2 cos 2x sin3x 13. y = ( x+3 2 x+2 32 14. y = 2x2. e3x+x² 15. y = (3x² + 5). In 2x² 16. у = 2e2x. In 2x4 найти у (1)

y = 8x²⁵

y' = 8 \cdot 25x^25-1 = 200x²⁴

y = 15x^-3

y' = 15 \cdot (-3)x^-3-1 = -45x^-4 = -\frac{45}{x^4}

y = 12x³ - 2x + 14

y' = 12 \cdot 3x² - 2 = 36x² - 2

y = \frac{-2-7x}{1-3x}

y' = \frac{(-7(1-3x) - (-2-7x)(-3))}{(1-3x)^2} = \frac{-7 + 21x - 6 - 21x}{(1-3x)^2} = \frac{-13}{(1-3x)^2}

y = (3x⁵ + 7x² + 10x - 7)⁴

y' = 4(3x⁵ + 7x² + 10x - 7)³ \cdot (15x⁴ + 14x + 10)

y = \frac{1}{x^6} = x^{-6}

y' = -6x^{-7} = -\frac{6}{x^7}

y'(1) = -\frac{6}{1^7} = -6

y = \sqrt{2x^3 - 4} = (2x^3 - 4)^{1/2}

y' = \frac{1}{2}(2x^3 - 4)^{-1/2} \cdot (6x^2) = \frac{3x^2}{\sqrt{2x^3 - 4}}

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

y' = \sqrt{2} (\frac{1}{2}x^{-\frac{1}{2}}(3x+1) + x^{\frac{1}{2}} \cdot 3) = \sqrt{2} (\frac{3x+1}{2\sqrt{x}} + 3\sqrt{x}) = \sqrt{2} (\frac{3x+1+6x}{2\sqrt{x}}) = \sqrt{2} \cdot \frac{9x+1}{2\sqrt{x}}

y = 7^4x+1

y' = 7^{4x+1} \cdot ln(7) \cdot 4 = 4 \cdot 7^{4x+1} \cdot ln(7)

y = ln(2x^3 + 6)

y' = \frac{6x^2}{2x^3 + 6} = \frac{3x^2}{x^3 + 3}

y = ctg^3(x) + cos(2x^4)

y' = 3ctg^2(x) \cdot (-1 - ctg^2(x)) - sin(2x^4) \cdot 8x^3 = -3ctg^2(x)(1 + ctg^2(x)) - 8x^3sin(2x^4)

y = 2cos(2x) \cdot sin(3x)

y' = 2(-2sin(2x)sin(3x) + cos(2x)3cos(3x)) = -4sin(2x)sin(3x) + 6cos(2x)cos(3x)

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

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

y = 2x^2 \cdot e^{3x+x^2}

y' = 4x \cdot e^{3x+x^2} + 2x^2 \cdot e^{3x+x^2} \cdot (3 + 2x) = e^{3x+x^2} (4x + 6x^2 + 4x^3)

y = (3x^2 + 5) \cdot ln(2x^2)

y' = 6x \cdot ln(2x^2) + (3x^2 + 5) \cdot \frac{4x}{2x^2} = 6x \cdot ln(2x^2) + \frac{2(3x^2 + 5)}{x}

y = 2e^{2x} \cdot ln(2x^4)

y' = 4e^{2x}ln(2x^4) + 2e^{2x} \cdot \frac{8x^3}{2x^4} = 4e^{2x}ln(2x^4) + \frac{8e^{2x}}{x}

y'(1) = 4e^{2}ln(2) + 8e^{2}