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.

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

Answer 2

1. y = -6x² + 15x² - 36x + 20

To find the derivative of the function y = -6x² + 15x² - 36x + 20, we can differentiate each term separately using the power rule. The power rule states that if we have a term of the form ax^n, the derivative is given by multiplying the coefficient a by the exponent n and then subtracting 1 from the exponent.

Let's differentiate each term step by step:

- The derivative of -6x² is -12x. (Using the power rule, we multiply the coefficient -6 by the exponent 2 and subtract 1 from the exponent.) - The derivative of 15x² is 30x. (Similarly, we multiply the coefficient 15 by the exponent 2 and subtract 1 from the exponent.) - The derivative of -36x is -36. (Here, the exponent is 1, so we multiply the coefficient -36 by 1 and subtract 1 from the exponent, which gives us 0. Any term with an exponent of 0 becomes a constant, and the derivative of a constant is 0.) - The derivative of 20 is 0