Найти производную функции y = 2x^3+5/√x^4+2x

6x² + 6/x² + 5x + 5/x - 38 = 0

6(x² + 1/x²) + 5(1/x + x) - 38 = 0

x ≠ 0

замена

1/x + x = t

(1/x + x)² = t²

1/x² + 2*1/x * x + x² = t²

1/x² + 2 + x² = t²

1/x² + x² = t² - 2

6(x² + 1/x²) + 5(1/x + x) - 38 = 0

6(t² - 2) + 5t - 38 = 0

6t² - 12 + 5t - 38 = 0

6t² + 5t - 50 = 0

D = 25 + 4*50*6 = 1225 = 35²

t12 = (-5 +- 35)/12 = 30/12 (5/2)   - 40/12 (-10/3)

обратно к х

1. 1/x + x = 5/2

2x² - 5x + 2 = 0

D = 25 - 16 = 9 = 3²

x12 = (5 +- 3)/4 = 2    1/2

2. 1/x + x = -10/3

3x² + 10x + 3 = 0

D = 100 - 36 = 64 = 8²

x12 = (-10 +- 8)/6 = -3  -1/3

ответ x = {2,1/2,-3,-1/3}

вкратце

1) sin x = √2/2

  x = (-1)ⁿ × arcsin √2/2 + πn, n∈Z

  x = (-1)ⁿ × π/4 + πn, n∈Z

2) sin x = -√2/2

   x = (-1)ⁿ × arcsin (-√2/2) + πn, n∈Z

   x = (-1)ⁿ × -arcsin √2/2 + πn, n∈Z

   x = (-1)ⁿ × (-π/4) + πn, n∈Z

3) sin x = -√3/2

   x = (-1)ⁿ × arcsin (-√3/2) + πn, n∈Z

   x = (-1)ⁿ × -arcsin √3/2 + πn, n∈Z

   x = (-1)ⁿ × (-π/3) + πn, n∈Z

4) sin x = √3/2

   x = (-1)ⁿ × arcsin √3/2 + πn, n∈Z

   x = (-1)ⁿ × arcsin √3/2 + πn, n∈Z

   x = (-1)ⁿ × π/3 + πn, n∈Z

5) sin x = 4/5

   x = (-1)ⁿ × arcsin 4/5 + πn, n∈Z

   x = (-1)ⁿ × 0,927295 + πn, n∈Z

   x = (-1)ⁿ × 53,1° + πn, n∈Z