Question

Вычисление производных.

$lg ^{2} sqrt{x}$
$ln (x+ sqrt{1+ x^{2} } )$
$log_5 sqrt{2 x^{2} -3x+4}$

Answer 1

$y=lg^2sqrt{x}\\y'=2lgsqrt{x}cdot frac{1}{sqrt{x}ln10}cdot frac{1}{2sqrt{x}}=frac{lgsqrt{x}}{xln10}\\\y=ln(x+sqrt{1+x^2})\\y'=frac{1}{x+sqrt{1+x^2}}cdot (1+frac{1}{2sqrt{1+x^2}}cdot 2x)}=frac{1}{x+sqrt{1+x^2}}cdot frac{sqrt{1+x^2}+x}{sqrt{1+x^2}}=frac{1}{sqrt{1+x^2}}\\\y=log_5sqrt{2x^2-3x+4}\\y'=frac{1}{sqrt{2x^2-3x+4}, cdot ln5}cdot frac{1}{2sqrt{2x^2-3x+4}}cdot (4x-3)=frac{4x-3}{2ln5(2x^2-3x+4)}$