Question

Найдите производные следующих функций: 1)y=x^4/√x
2)y=2x^3/√x^3
3)y=2√x/x^3
4)y=6*3√x/x
5)y=4√1/x^3​

Answer 1

1.

$y = \frac{ {x}^{4} }{ \sqrt{x} } = {x}^{3 \frac{1}{2} } = {x}^{ \frac{7}{2} } \\$

$y' = \frac{7}{2} {x}^{ \frac{5}{2} } = 3.5 {x}^{2} \sqrt{x} \\$

2.

$y = \frac{2 {x}^{3} }{ \sqrt{ {x}^{3} } } = 2 {x}^{3 - \frac{3}{ 2} } = 2 {x}^{ \frac{3}{2} } \\$

$y '= 2 \times \frac{3}{2} {x}^{ \frac{1}{2} } = 3 \sqrt{x} \\$

3.

$y = \frac{2 \sqrt{x} }{ {x}^{3} } = 2 {x}^{ \frac{1}{2} - 3} = 2 {x}^{ - \frac{5}{2} } \\$

$y' = 2 \times ( - \frac{5}{2} ) {x}^{ - \frac{7}{2} } = - \frac{5}{ {x}^{3} \sqrt{x} } \\$

4.

$y = \frac{6 \times 3 \sqrt{x} }{x} = 18 {x}^{ - \frac{1}{2} } \\$

$y' = 18 \times ( - \frac{1}{2} ) {x}^{ - \frac{3}{2} } = - \frac{9}{x \sqrt{x} } \\$

5.

$y = \frac{4}{ \sqrt{ {x}^{3} } } = 4 {x}^{ - \frac{3}{2} } \\$

$y' = 4 \times ( - \frac{3}{2} ) {x}^{ - \frac{5}{2} } = - \frac{6}{ {x}^{2} \sqrt{x} } \\$