22.6. EXERCISES 477
(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 + x2x2
2 − x1 + x2x2
2 + x1 + sinx2
.
Find D( f ◦g)(x). Use to write ∂ z∂xi
for i = 1,2.
3. Let z = f (y) =(y2
1 + coty2 + siny3)
and y = g (x) ≡
x1 + x4 + x3x2
2 − x1 + x2x2
2 + x1 + sinx4
. Find
D( f ◦g)(x). Use to write ∂ z∂xi
for i = 1,2,3,4.
4. Let z = f (y) =(y2
1 + y22 + siny3 + y4
), y = g (x)≡
x1 + x4 + x3x2
2 − x1 + x2x2
2 + x1 + sinx4x4 + x2
. Find the
derivative of the composition D( f ◦g)(x). Use to write ∂ z∂xi
for i = 1,2,3,4.
5. Let
z = f (y) =
(y2
1 + siny2 + tany3y2
1y2 + y3
)
and y = g (x) ≡
x1 + x2x2
2 − x1 + x2x2
2 + x1 + sinx2
. Find D(f ◦g)(x). Use to write ∂ zk∂xi
for
i = 1,2 and k = 1,2. Recall this will be of the form(
z1x1 z1x2 z1x3z2x1 z2x2 z2x3
).
6. Let z = f (y) =
y21 + siny2 + tany3
y21y2 + y3
cos(y2
1)+ y3
2y3
and
y = g (x)≡
x1 + x4x2
2 − x1 + x3x2
3 + x1 + sinx2
.
Find D(f ◦g)(x). Use to write ∂ zk∂xi
for i = 1,2,3,4 and k = 1,2,3.
7. Give a version of the chain rule which involves three functions f,g,h.
8. If f :U →V and f−1 : V →U for U,V open sets such that f,f−1 are both differen-tiable, show that
det(Df(f−1 (y)
))det(Df−1 (y)
)= 1