Question

Решить производные функции.

Примеры:
1. y=-6x²+15x²-36x+20
2. y=(19x²+9x+11)^6
3. y=8^10x³+x⁴ (После знака ^ все под степенью)
4. y=ln(7x⁴+3x³+1)
5. y=cos(24x²-18x)
6. y=(x²+2x)(4x+5)

Распишите подробнее, пожалуйста.

Answer 1

Ответ:

1.

$y' = - 6 \times 2x + 15 \times 2x - 36 + 0 = \\ = - 12x + 30x - 36 = 18x - 36$

2.

$y' = 6 {(19 {x}^{2} + 9x + 11) }^{5} \times (19 {x}^{2} + 9x + 11)' = \\ = 6 {(19 {x}^{2} + 9x + 11)}^{5} \times (19 \times 2x + 9 + 0) = \\ = 6 {(19 {x}^{2} + 9x + 11) }^{5} (38x + 9)$

3.

$y' = ln(8) \times {8}^{10 {x}^{3} + {x}^{4} } \times (10 {x}^{3} + {x}^{4} )' = \\ = ln(8) \times {8}^{10 {x}^{3} + {x}^{4} } \times (30 {x}^{2} + 4 {x}^{3} )$

4.

$y' = \frac{1}{7 {x}^{4} + 3 {x}^{3} + 1} \times (7 {x}^{4} + 3 {x}^{3} + 1)' = \\ = \frac{28 {x}^{3} + 9 {x}^{2} }{7 {x}^{4} + 3 {x}^{3} + 1}$

5.

$y' = - \sin(24 {x}^{2} - 18x) \times (24 {x}^{2} - 18x) '= \\ = - \sin(24 {x}^{2} - 18x ) \times (48x - 18)$

6.

$y' = ( {x}^{2} + 2x)'(4x + 5) + (4x + 5)'( {x}^{2} + 2x) = \\ = (2x + 2)(4x + 5) + 4( {x}^{2} + 2x) = \\ = 8 {x}^{2} + 10x + 8x + 10 + 4 {x}^{2} + 8x = \\ = 12 {x}^{2} + 26x + 10$