4.2 Yhdistetyn funktion derivaatta

421

a)
$f\left(x\right)=\left(x^2+3\right)^{11}$

$s\left(x\right)=x^2+3$
$u\left(x\right)=x^{11}$

b)
$f\left(x\right)=\left(x^2+3\right)^{11}$
$D\left(u\left(s\left(x\right)\right)\right)=u'\left(s\left(x\right)\right)s'\left(x\right)$

$11\cdot\left(x^2+3\right)^{10}\cdot2x=22x\left(x^2+3\right)^{10}$

422

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

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

$u\left(x\right)=x^4$ , $u'\left(x\right)=4x^3$

$D\left(u\left(s\left(x\right)\right)\right)=u'\left(s\left(x\right)\right)s'\left(x\right)$

$4\cdot\left(4x+2\right)^3\cdot4=16\cdot\left(4x+2\right)^3$

b) $f\left(x\right)=\sin2x$

$s\left(x\right)=2x$ , $s'\left(x\right)=2$

$u\left(x\right)=\sin x$ , $u'\left(x\right)=\cos x$

$D\left(u\left(s\left(x\right)\right)\right)=u'\left(s\left(x\right)\right)s'\left(x\right)$

$\cos\cdot\left(2x\right)\cdot2=2\cos2x$

c) $f\left(x\right)=\cos3x$

$s\left(x\right)=3x$ , $s'\left(x\right)=3$

$u\left(x\right)=\cos x$ , $u'\left(x\right)=-\sin x$

$D\left(u\left(s\left(x\right)\right)\right)=u'\left(s\left(x\right)\right)s'\left(x\right)$
$-\sin\left(3x\right)\cdot3=-3\sin3x$

423

$D\left(u\left(s\left(x\right)\right)\right)=u'\left(s\left(x\right)\right)s'\left(x\right)$

a) $D\left(-2x+4\right)^6$

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

$u\left(x\right)=x^6$ , $u'\left(x\right)=6x^5$

$6\left(-2x+4\right)^5\cdot\left(-2\right)=-12\left(-2x+4\right)^5$

b) $D\left(x^2-1\right)^5$

$s\left(x\right)=x^2-1$ , $s'\left(x\right)=2x$

$u\left(x\right)=x^5$ , $u'\left(x\right)=5x^4$

$5\left(x^2-1\right)^4\cdot2x=10x\left(x^2-1\right)^4$

c) $D\sin x^2$

$D\left(u\left(s\left(x\right)\right)\right)=u'\left(s\left(x\right)\right)s'\left(x\right)$

$s\left(x\right)=x^2$ , $s'\left(x\right)=2x$
$u\left(x\right)=\sin x$ , $u'\left(x\right)=\cos x$

$\cos\left(x^2\right)\cdot2x=2x\cdot\cos x^2$

d) $D\cos^2x=D\ \left(\cos x\right)^2$

$s\left(x\right)=\cos x$ , $s'\left(x\right)=-\sin x$

$u\left(x\right)=x^2$ , $u'\left(x\right)=2x$

$2\left(\cos x\right)\cdot\left(-\sin x\right)=-2\sin x\cdot\left(\cos x\right)$