Question

Знайти похідну функції:

Answer 1

1.

$y = 24 \\ y' = 0 \\ y = {x}^{16} \\ y' = 16 {x}^{16 - 1} = 16 {x}^{15} \\ y = {x}^{ - 7} \\ y' = - 7 {x}^{ - 7 - 1} = - 7 {x}^{ - 8} = - \frac{7}{ {x}^{8} }$

2.

$y = 8 {x}^{3} - {x}^{2} + 6 \\ y' = 3 \times 8 {x}^{3 - 1} - 2 {x}^{2 - 1} = 24 {x}^{2} - 2x$

3.

$f(x) = \sin(x) - \cos(x) \\ f'(x) = \cos(x) - ( - \sin(x) ) = \cos(x) + \sin(x)$

4.

$y = \frac{3x - 1}{3x + 1} \\ y' = \frac{(3x - 1)'(3x + 1) - (3x + 1)'(3x - 1)}{(3x + 1) {}^{2} } = \\ = \frac{3(3x + 1) - 3(3x - 1)}{(3x + 1) {}^{2} } = \frac{3(3x + 1 - 3x + 1)}{(3x + 1) {}^{2} } = \\ \frac{3 \times2 }{(3x + 1) {}^{2} } = \frac{6}{(3x + 1) {}^{2} }$

5.

$y = (1 - 3x) {}^{5} \\ y' = - 3 \times 5(1 - 3x) {}^{5 - 1} = - 15(1 - 3x) {}^{4}$

6.

$y = \cos(3x) \\ y' = - 3 \sin(3x)$