Question

Найдите f"(x) вторая производная функции

Answer 1

$1)f'(x) = \frac{1}{2} {x}^{ - \frac{1}{2} } = \frac{1}{2 \sqrt{x} } \\ f''(x) = \frac{1}{2} \times ( - \frac{1}{2} ) {x}^{ - \frac{3}{2} } = \\ = - \frac{1}{4x \sqrt{x} }$

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

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

$f''(x) = - \frac{1}{2} \times ( - \frac{1}{2} ) {( - x)}^{ - \frac{3}{2} } \times ( - 1) = \\ = - \frac{1}{4 \sqrt{(-x)^{3}} }$

$4)f'(x) = ( {x}^{ \frac{3}{2} } )' = \frac{3}{2} {x}^{ \frac{1}{2} } = \\ = 1.5 \sqrt{x}$

$f''(x) = \frac{3}{2} \times \frac{1}{2} {x}^{ -\frac{1}{2} } = \frac{3}{4\sqrt{x}}$