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)