oblicz pierwsza i druga pochodna

[olciaa]

\(f(x)= \frac{1}{2}x \sqrt{2-x^2}+arcsin \frac{x}{ \sqrt{2} }\)

[jola]

\(f'(x)= \frac{1}{2} \sqrt{2-x^2} + \frac{1}{2} x \cdot \frac{-2x}{2 \sqrt{2-x^2} }+ \frac{ \frac{1}{ \sqrt{2} } }{ \sqrt{1- \frac{x^2}{2} } } = \frac{2-x^2-x^2}{2 \sqrt{2-x^2} }+ \frac{1}{ \sqrt{2-x^2} }= \frac{1-x^2}{ \sqrt{2-x^2} }+ \frac{1}{ \sqrt{2-x^2} }= \sqrt{2-x^2}\)

\(f''(x)= \frac{-2x}{2 \sqrt{2-x^2} }= \frac{-x}{ \sqrt{2-x^2} }\)