Question

Помогите найти первую производную функции.

Answer 1

$y=lnleft(x+sqrt{a^2+x^2}right)-2sqrt{x^2+a^2} medskip \ y'=left[lnleft(x+sqrt{x^2+a^2}right)right]'-2left(sqrt{x^2+a^2}right)'=medskip\=dfrac{1}{x+sqrt{x^2+a^2}}cdot(x+sqrt{x^2+a^2})'-dfrac{2cdot 2x}{2sqrt{x^2+a^2}}=medskip\=dfrac{1+dfrac{2x}{2sqrt{x^2+a^2}}}{x+sqrt{x^2+a^2}}-dfrac{2x}{sqrt{x^2+a^2}}=dfrac{1+dfrac{x}{sqrt{x^2+a^2}}}{x+sqrt{x^2+a^2}}-dfrac{2x}{sqrt{x^2+a^2}}=$

$=dfrac{dfrac{x+sqrt{x^2+a^2}}{sqrt{x^2+a^2}}}{x+sqrt{x^2+a^2}}-dfrac{2x}{sqrt{x^2+a^2}}=dfrac{1-2x}{sqrt{x^2+a^2}}$

Ответ. $y'=dfrac{1-2x}{sqrt{x^2+a^2}}$