Найдите производные функций (208-211). 208.- a) f (x) = x² + x³; B) f (x) = x² + 3x -1; 209. a) f (x) = x3 (4+2x-x²); B) f (x) = x² (3x + x³); 210. a) y = 1+2x; б) y = x²; 3-5x 2x-1 211. a) y = x³-3x1-x+5; 6) y=x-4+√x; B) y = x² - 4x5 + 2x - 1: n 23 6) f (x)= =+5x-2; X r) f (x) = x² + √x. 6) f(x) = x(2x²-x); r) f (x) = (2x-3) (1-x). 3x-2 5x+8 3-4x -; B) y =; г) у= x2 B) y = x² - 4x5 + 2x - 1: r) u=+=+1.

208.

1. a) f(x) = x² + x³

   Производная суммы равна сумме производных:

   f'(x) = (x²)’ + (x³)’ = 2x + 3x²

   f'(x) = 2x + 3x²

2. в) f(x) = x² + 3x - 1

   f'(x) = (x²)’ + (3x)’ - (1)’ = 2x + 3 - 0

   f'(x) = 2x + 3

3. б) f(x) = 1/x + 5x - 2

   f'(x) = (1/x)’ + (5x)’ - (2)’ = -1/x² + 5 - 0

   f'(x) = -1/x² + 5

4. г) f(x) = x³ + √x

   f'(x) = (x³)’ + (√x)’ = 3x² + 1/(2√x)

   f'(x) = 3x² + 1/(2√x)

209.

1. a) f(x) = x³(4 + 2x - x²)

   f(x) = 4x³ + 2x⁴ - x⁵

   f'(x) = (4x³)’ + (2x⁴)’ - (x⁵)’ = 12x² + 8x³ - 5x⁴

   f'(x) = 12x² + 8x³ - 5x⁴

2. в) f(x) = x²(3x + x³)

   f(x) = 3x³ + x⁵

   f'(x) = (3x³)’ + (x⁵)’ = 9x² + 5x⁴

   f'(x) = 9x² + 5x⁴

3. б) f(x) = √x(2x² - x)

   f(x) = 2x^(5/2) - x^(3/2)

   f'(x) = (2x^(5/2))’ - (x^(3/2))’ = 5x^(3/2) - (3/2)x^(1/2)

   f'(x) = 5x^(3/2) - (3/2)x^(1/2)

4. г) f(x) = (2x - 3)(1 - x³)

   f(x) = 2x - 2x⁴ - 3 + 3x³

   f'(x) = (2x)’ - (2x⁴)’ - (3)’ + (3x³)’ = 2 - 8x³ - 0 + 9x²

   f'(x) = 2 - 8x³ + 9x²

210.

1. a) y = (1 + 2x) / (3 - 5x)

   y' = ((1 + 2x)' * (3 - 5x) - (1 + 2x) * (3 - 5x)') / (3 - 5x)²

   y' = (2 * (3 - 5x) - (1 + 2x) * (-5)) / (3 - 5x)²

   y' = (6 - 10x + 5 + 10x) / (3 - 5x)²

   y' = 11 / (3 - 5x)²

2. б) y = x² / (2x - 1)

   y' = ((x²)' * (2x - 1) - x² * (2x - 1)') / (2x - 1)²

   y' = (2x * (2x - 1) - x² * 2) / (2x - 1)²

   y' = (4x² - 2x - 2x²) / (2x - 1)²

   y' = (2x² - 2x) / (2x - 1)²

3. в) y = (3x - 2) / (5x + 8)

   y' = ((3x - 2)' * (5x + 8) - (3x - 2) * (5x + 8)') / (5x + 8)²

   y' = (3 * (5x + 8) - (3x - 2) * 5) / (5x + 8)²

   y' = (15x + 24 - 15x + 10) / (5x + 8)²

   y' = 34 / (5x + 8)²

4. г) y = (3 - 4x) / x²

   y' = ((3 - 4x)' * x² - (3 - 4x) * (x²)') / x⁴

   y' = (-4 * x² - (3 - 4x) * 2x) / x⁴

   y' = (-4x² - 6x + 8x²) / x⁴

   y' = (4x² - 6x) / x⁴

   y' = (4x - 6) / x³

211.

1. a) y = x⁸ - 3x⁴ - x + 5

   y' = (x⁸)’ - (3x⁴)’ - (x)’ + (5)’ = 8x⁷ - 12x³ - 1 + 0

   y' = 8x⁷ - 12x³ - 1

2. в) u = x⁷ - 4x⁵ + 2x - 1

   u' = (x⁷)’ - (4x⁵)’ + (2x)’ - (1)’ = 7x⁶ - 20x⁴ + 2 - 0

   u' = 7x⁶ - 20x⁴ + 2

3. б) y = x/3 - 4/x² + √x

   y' = (x/3)’ - (4/x²)’ + (√x)’ = 1/3 + 8/x³ + 1/(2√x)

   y' = 1/3 + 8/x³ + 1/(2√x)

4. г) u = x²/3 + 3/x² + 1

   u' = (x²/3)’ + (3/x²)’ + (1)’ = (2/3)x - 6/x³ + 0

   u' = (2/3)x - 6/x³