17.7. EXERCISES 315
where each ui is given by the above formulas. Thus ∂ z1∂x1
equals
2u1−u3 = 2(x1 + x2
2 + sin(x3)+ cos(x4))−(x2
3 + x4)
= 2x1 +2x22 +2sinx3 +2cosx4− x2
3− x4.
while ∂ z2∂x4
equals
−2u1 sinx4 +6u22x4 =−2
(x1 + x2
2 + sin(x3)+ cos(x4))
sin(x4)+6(x2
4− x1)2
x4.
If you wanted ∂z∂x2
it would be the second column of the above matrix in (17.12). Thus ∂z∂x2
equals ∂ z1∂x2∂ z2∂x2∂ z3∂x2
=
4u1x2
4u1x2
0
=
4(x1 + x2
2 + sin(x3)+ cos(x4))
x2
4(x1 + x2
2 + sin(x3)+ cos(x4))
x2
0
I hope that by now it is clear that all the information you could desire about various partialderivatives is available and it all reduces to matrix multiplication and the consideration ofentries of the matrix obtained by multiplying the two derivatives.
17.7 Exercises1. Let z = f (x1, · · · ,xn) be as given and let xi = gi (t1, · · · , tm) as given. Find ∂ z
∂ tiwhich
is indicated.
(a) z = x31 + x2, x1 = sin(t1)+ cos(t2) ,x2 = t1t2
2 . Find ∂ z∂ t1
(b) z = x1x22, x1 = t1t2
2 t3,x2 = t1t22 . Find ∂ z
∂ t1.
(c) z = x1x22, x1 = t1t2
2 t3,x2 = t1t22 . Find ∂ z
∂ t1.
(d) z = x1x22, x1 = t1t2
2 t3,x2 = t1t22 . Find ∂ z
∂ t3.
(e) z = x21x2
2, x1 = t1t22 t3,x2 = t1t2
2 . Find ∂ z∂ t2
.
(f) z = x21x2 + x2
3, x1 = t1t2,x2 = t1t2t4,x3 = sin(t3). Find ∂ z∂ t2
.
(g) z = x21x2 + x2
3, x1 = t1t2,x2 = t1t2t4,x3 = sin(t3). Find ∂ z∂ t3
.
(h) z = x21x2 + x2
3, x1 = t1t2,x2 = t1t2t4,x3 = sin(t3). Find ∂ z∂ t1
.
2. Let z = f (y) =(y2
1 + siny2 + tany3)
and
y = g (x)≡
x1 + x2
x22− x1 + x2
x22 + x1 + sinx2
.
Find D( f ◦g)(x). Use to write ∂ z∂xi
for i = 1,2.