474 CHAPTER 22. DERIVATIVE OF A FUNCTIONS OF MANY VARIABLES
Example 22.5.5 Let w = f (u1,u2,u3) = u21 +u3 +u2 and
g (x,y,z) =
u1u2u3
=
x+2yzx2 + yz2 + x
Find ∂w
∂x and ∂w∂ z
By the chain rule,
(wx,wy,wz) =(
wu1 wu2 wu3
) u1x u1y u1zu2x u2y u2zu3x u3y u3z
=
(wu1u1x +wu2u2x +wu3u3x,wu1u1y +wu2u2y +wu3u3y,
wu1u1z +wu2u2z +wu3u3z)
Note the pattern,
wx = wu1u1x +wu2u2x +wu3u3x,
wy = wu1u1y +wu2u2y +wu3u3y,
wz = wu1u1z +wu2u2z +wu3u3z.
Therefore,
wx = 2u1 (1)+1(2x)+1(1) = 2(x+2yz)+2x+1 = 4x+4yz+1
andwz = 2u1 (2y)+1(0)+1(2z) = 4(x+2yz)y+2z = 4yx+8y2z+2z.
Of course to find all the partial derivatives at once, you just use the chain rule. Thus youwould get
(wx wy wz
)=(
2u1 1 1) 1 2z 2y
2x 1 01 0 2z
=
(2u1 +2x+1 4u1z+1 4u1y+2z
)=
(4x+4yz+1 4zx+8yz2 +1 4yx+8y2z+2z
)Example 22.5.6 Let f (u1,u2) =
(u2
1 +u2sin(u2)+u1
)and
g (x1,x2,x3) =
(u1 (x1,x2,x3)u2 (x1,x2,x3)
)=
(x1x2 + x3x2
2 + x1
).
Find D(f ◦g)(x1,x2,x3).