Найдите производную 20 баллов...

vipsergazyev @vipsergazyev

August 2022 1 18 Report

Найдите производную
20 баллов

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sangers1959

Verified answer

Ответ:

Объяснение:

$y=\frac{\sqrt{a^{2}+x^{2} } }{x} \\y'=(\frac{\sqrt{a^{2}+x^{2} } }{x})'=\frac{(\sqrt{a^{2}+x^{2} } )'*x-\sqrt{a^{2}+x^{2} } *x'}{x^{2} }$

Вычислим числитель:

$\sqrt{a^{2}+x^{2} } )'*x-\sqrt{a^{2}+x^{2} } *x'=\frac{x*(a^{2}+x^{2})' }{2*\sqrt{a^{2}+x^{2} } }-\sqrt{a^{2}+x^{2} }=\frac{x*2x-2*(\sqrt{a^{2} +x^{2} } )^{2} }{2*\sqrt{a^{2}+x^{2} } } =\\ =\frac{2x^{2}-2a^{2}+2x^{2} }{2*\sqrt{a^{2}+x^{2} } } =\frac{-2a^{2} }{2*\sqrt{a^{2} +x^{2} } } =-\frac{a^{2} }{\sqrt{a^{2} +x^{2} } } .$

Таким образом:

$y'=-\frac{a^{2} }{x^{2}*\sqrt{a^{2}+x^{2} } } .$

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