Question

Calculati derivata functiei:
$f_{(x)}= arcsin \frac{2x}{1+ x^{2} }$

Answer 1

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