Question

Знайти похідні заданих функцій:

Answer 1

Ответ:

3.

$y' = (2 {x}^{2} + 3)'( {x}^{3} + 2x + 5) + ( {x}^{3} + 2x + 5)'(2 {x}^{2} + 3) = \\ = 4x( {x}^{3} + 2x + 5) + (3 {x}^{2} + 2)(2 {x}^{2} + 3) = \\ = 4 {x}^{4} + 8 {x}^{2} + 20x + 6 {x}^{4} + 9 {x}^{2} + 4 {x}^{2} + 6 = \\ = 10 {x}^{4} + 21 {x}^{2} + 20x + 6$

$y(2) = 10 \times 16 + 21 \times 4 + 20 \times 2 + 6 = \\ = 160 + 84 + 46 = 290$

4.

$y' = \frac{( {x}^{4} + 1)'({x}^{3} + 2) - ( {x}^{3} + 2)'( {x}^{4} + 1) }{ {( {x}^{3} + 2)}^{2} } = \\ = \frac{4 {x}^{3} ( {x}^{3} + 2) - 3 {x}^{2} ( {x}^{4} + 1)}{ {( {x}^{3} + 2)}^{2} } = \\ = \frac{4 {x}^{6} + 8 {x}^{3} - 3 {x}^{6} - 3 {x}^{2} }{ {( {x}^{3} + 2)}^{2} } = \frac{ {x}^{6} + 8 {x}^{3} - 3 {x}^{2} }{ {( {x}^{3} + 2)}^{2} }$

5.

$y '= \cos(x) - \sin(x)$

6.

$y' = ( {x}^{2} - 2x + 3)'e {}^{x} + ( e {}^{x})' ( {x}^{2} - 2x + 3) = \\ = (2x - 2) {e}^{x} + e {}^{x} ( {x}^{2} - 2 x+ 3) = \\ = {e}^{x} ( {x}^{2} + 1)$

7.

$y' = 3 {x}^{2} - ln(3) \times {3}^{x}$

8.

$y '= 2x log_{3}(x) + \frac{ {x}^{2} }{ ln( 3) \times x } = \\ = 2x log_{ 3}(x) + \frac{x}{ ln(3) }$

9.

$f'(x) = 3 - 2 \times \frac{1}{2} {x}^{ - \frac{1}{2} } = 3 - \frac{1}{ \sqrt{x} }$