Question

Найти вторую производную функции Y:

Answer 1

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

$y'' = \frac{(2 - ln(x - 1)) '\times 2 \sqrt{ {(x - 1)}^{3} } - (2 {(x - 1)}^{ \frac{3}{2} } ) '\times (2 - ln(x - 1)) }{4 {(x - 1)}^{3} } = \\ = \frac{ - \frac{1}{x - 1} \times 2 \sqrt{ {(x - 1)}^{3} } - 2 \times \frac{3}{2} \sqrt{x - 1} \times (2 - ln(x - 1)) }{4 {(x - 1)}^{3} } = \\ = \frac{ - 2 \sqrt{x - 1} - 3 \sqrt{x - 1}(2 - ln(x - 1)) }{4 {(x - 1)}^{3} } = \\ = - \frac{ \sqrt{x - 1} (2 + 3(2 - ln(x - 1))) }{4 {(x - 1)}^{3} } = \\ = - \frac{2 + 6 - 3 ln(x - 1) }{4 \sqrt{ {(x - 1)}^{5} } } = - \frac{8 - 3 ln(x - 1) }{4 \sqrt{ {(x - 1)}^{5} } } = \\ = \frac{3 ln(x - 1) - 8 }{4 \sqrt{ {(x - 1)}^{5} } }$