Question

Найти производную функции. Задание на картинке.

Answer 1

Ответ:

Найти производную дроби .

Формулы :    $\bf \Big(\dfrac{u}{v}\Big)'=\dfrac{u'v-uv'}{v^2}\ \ \ ,\ \ \ \ (\sqrt{u})'=\dfrac{u'}{2\sqrt{u}}$   .  

$\bf \displaystyle y=\frac{\sqrt{1-x^2}+1}{\sqrt{1-x^2}-1}\\\\\\y'=\frac{({\sqrt{1-x^2}+1)'(\sqrt{1-x^2}-1)}-(\sqrt{1-x^2}+1)(\sqrt{1-x^2}-1)'}{(\sqrt{1-x^2}-1)^2}=\\\\\\=\frac{\dfrac{-2x}{2\sqrt{1-x^2}}\cdot (\sqrt{1-x^2}-1)-(\sqrt{1-x^2}+1)\cdot \dfrac{-2x}{2\sqrt{1-x^2}}}{(\sqrt{1-x^2}-1)^2}=\\\\\\=\frac{-x\cdot (\ \sqrt{1-x^2}-1-\sqrt{1-x^2}-1\ )}{\sqrt{1-x^2}\cdot (\sqrt{1-x^2}-1)^2}=\frac{2x}{\sqrt{1-x^2}\cdot (\sqrt{1-x^2}-1)^2}$