Kpl.4.2

$Du\left(s\left(x\right)\right)=u'\left(s\left(x\right)\right)\cdot s'\left(x\right)$

Esim. Derivoi

$f\left(x\right)=\left(2x+4\right)^5$

$s\left(x\right)=2x+4$

$u\left(x\right)=x^5$
$s'\left(x\right)=2{,}\ u'\left(x\right)=5x^4$

$u'\left(s\left(x\right)\right)=5x\left(2x+4\right)^4\cdot2=10x\left(2x+4\right)^4$

$p\left(x\right)=2x\cdot\tan\left(3x\right)$
$Df\left(x\right)g\left(x\right)=f'\left(x\right)g\left(x\right)+g'\left(x\right)f\left(x\right)$

$f\left(x\right)=2x$
$g\left(x\right)=\tan\left(3x\right)$

$s'\left(x\right)=3$
$u'\left(x\right)=\frac{1}{\cos^2x}\ tai\ u'\left(x\right)=1+\tan^2x$
$g'\left(x\right)=\frac{3}{\cos^2\left(3x\right)}\ \ tai\ g'\left(x\right)=3+3\tan^2\left(3x\right)$

$p'\left(x\right)=2\cdot\tan\left(3x\right)+2x\cdot\left(3+3\tan^2\left(3x\right)\right)=2\tan\left(3x\right)+6x+6x\tan^2\left(3x\right)$

$r\left(x\right)=\frac{\left(3x^2+2x\right)^3}{2x}$

$D\frac{f\left(x\right)}{g\left(x\right)}=\frac{f'\left(x\right)\cdot g\left(x\right)-g'\left(x\right)\cdot f\left(x\right)}{\left(g\left(x\right)\right)^2}$

$f'\left(x\right)=3\left(3x^2+2x\right)^2\left(6x+2\right)$
$r'\left(x\right)=\frac{3\left(3x^2+2x\right)^2\left(6x+2\right)\cdot2x-2\cdot\left(3x^2+2x\right)^3}{\left(2x\right)^2}=\frac{2\left(3x^2+2x\right)^2\left(3x\left(6x+2\right)-\left(3x^2+2x\right)\right)}{4x^2}=\frac{\left(3x^2+2x\right)^2\left(18x^2+6x-3x^2-2x\right)}{2x^2}=\frac{\left(3x^2+2x\right)^2\left(15x^2+4x\right)}{2x^2}$