Question

Найти производную f(x)=ln (5-x^2/5+x^2), вычислить f'(x)

Answer 1

$f(x) = ln( \frac{5 - {x}^{2} }{5 + {x}^{2} } ) \\$

$f'(x) = \frac{1}{ \frac{5 - {x}^{2} }{5 + {x}^{2} } } \times ( \frac{5 - {x}^{2} }{5 + {x}^{2} } )' = \\ = \frac{5 + {x}^{2} }{5 - {x}^{2} } \times \frac{(5 - {x}^{2}) '\times (5 + {x}^{2} ) - (5 + {x}^{2} )' \times (5 - {x}^{2} )}{ {(5 + {x}^{2} )}^{2} } = \\ = \frac{5 + {x}^{2} }{5 - {x}^{2} } \times \frac{ - 2x(5 + {x}^{2}) - 2x(5 - {x}^{2} )}{ {(5 + {x}^{2}) }^{2} } = \\ = \frac{ - 10x - 2 {x}^{3} - 10x + 2 {x}^{3} }{(5 - {x}^{2} )(5 + {x}^{2} )} = \\ = - \frac{20x}{25 - {x}^{4} }$