Найти производную ф-ии: 1) f(x)=√x²+1/x (все в корне) 2) f(x)=3√e^4x+3 3) f(x)=...

1)

$\displaystyle f(x)= \sqrt{ \frac{x^2+1}{x}}$

$\displaystyle f`(x)=( \sqrt{ \frac{x^2+1}{x}})`=( \frac{x^2+1}{x})^{1/2}=$

$\displaystyle = \frac{1}{2} \frac{x^2+1}{x}^{-1/2}* \frac{2x*x-(x^2+1)}{x^2}= \frac{2x^2-x^2-1}{2 \sqrt{ \frac{x^2+1}{x}} }= \frac{x^2-1}{2 \sqrt{ \frac{x^2+1}{x}}}$

2)

$\displaystyle f(x)=3 \sqrt{e^{4x}+3}$

$\displaystyle f`(x)=3* \frac{1}{ 2\sqrt{e^{4x}+3}}*e^{4x}*4= \frac{6e^{4x}}{ \sqrt{e^{4x}+3}}$

3)
$\displaystyle f(x)=2sin^2x*cosx$

$\displaystyle f`(x)=2(sin^2x*cosx)`=2((sin^2x)`*cosx+(cosx)`*sin^2x)=$

$\displaystyle = 2(2sinx*cosx*cosx+(-sinx)*sin^2x)=2(2sinx*cos^2x-sin^3x)$

$\displaystyle=4sinx*cos^2x-2sin^3x$