10. 4. 9. ) 8. 3. 7. ) 6. 2. 5. 1. exercises Thus, D g ( x ) = 2[cos( f ( x ) · f ( x ))] f ( x ) D f ( x ). ∂ x ∂ x ··· ∂ f ∂ f ··· f ]and D f = . f = [ f ...... ∂ x ∂ x ··· ∂ f ∂ f ⎡⎢⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎥⎦ f ( x ) D f ( x ),whereweregard f asarowmatrix, ∂ x ∂ x

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Marsden-3620111 VC September 27, 2011 9:27 132

132 Differentiation

s o l u t i o n By the chain rule, Dg(x) = cos [ f (x) · f (x)]Dh(x), where h(x) = [ f (x) · f (x)] = f

12

(x) + · · · + f

m2

(x). Then

Dh(x) =

! ∂h

x

1

· · · ∂h

x

n

"

=

! 2f

1

f

1

∂x

1

+ · · · + 2f

m

f

m

x

1

· · · 2f

1

f

1

∂x

n

+ · · · + 2f

m

f

m

x

n

"

,

which can be written 2 f (x)Df (x), where we regard f as a row matrix,

f = [ f

1

· · · f

m

] and Df =

⎢ ⎢

⎢ ⎢

⎢ ⎣

f

1

∂x

1

· · · ∂ f

1

∂x

n

.. . .. .

f

m

∂x

1

· · · ∂ f

m

∂x

n

⎥ ⎥

⎥ ⎥

⎥ ⎦ .

Thus, Dg(x) = 2[cos ( f (x) · f (x))] f (x)Df (x).

exercises

1.

If f : U ⊂ Rn→ R is differentiable, prove that x#→ f2(x)+ 2 f (x) is differentiable as well, and compute its derivative in terms of Df (x).

2.

Prove that the following functions are differentiable, and find their derivatives at an arbitrary point:

(a) f :R2→ R, (x, y) #→ 2 (b) f :R2→ R, (x, y) #→ x + y (c) f :R2→ R, (x, y) #→ 2 + x + y (d) f :R2→ R, (x, y) #→ x2+ y2 (e) f :R2→ R, (x, y) #→ ex y (f ) f : U→ R, (x, y) #→

)

1− x2− y2, where U= {(x, y) | x2+ y2<1}

(g) f :R2→ R, (x, y) #→ x4− y4

3.

Verify the first special case of the chain rule for the composition f ◦ c in each of the cases:

(a) f (x, y)= xy, c(t) = (et, cos t) (b) f (x, y)= ex y, c(t)= (3t2, t3) (c) f (x, y)= (x2+ y2) log

)

x2+ y2, c(t)= (et, e−t)

(d) f (x, y)= x exp(x2+ y2), c(t)= (t, −t)

4.

What is the velocity vector for each path c(t) in

Exercise 3? [The solution to part (b) only is in the Study Guide to this text.]

5.

Let f :R3→ R and g: R3→ R be differentiable. Prove that

∇( f g) = f ∇g + g∇ f.

6.

Let f :R3→ R be differentiable. Making the substitution

x= ρ cos θ sin φ, y= ρ sin θ sin φ, z= ρ cos φ

(spherical coordinates) into f (x, y, z), compute

f /∂ρ, ∂ f /∂θ, and ∂ f /∂φ in terms of

f /∂ x, ∂ f /∂ y, and ∂ f /∂z.

7.

Let f (u, v)= (tan (u − 1) − ev, u2− v2) and g(x, y)= (ex−y, x− y). Calculate f ◦ g and D( f ◦ g)(1, 1).

8.

Let f (u, v, w)= (eu−w, cos (v + u) +

sin (u + v + w)) and g(x, y) = (ex, cos ( y− x), e−y).

Calculate f ◦ g and D( f ◦ g)(0, 0).

9.

Find (∂/∂s)( f ◦ T )(1, 0), where f (u, v) = cos u sin v and T :R2→ R2is defined by

T (s, t)= (cos (t2s), log√ 1+ s2).

10.

Suppose that the temperature at the point (x, y, z) in space is T (x, y, z)= x2+ y2+ z2. Let a particle follow

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Marsden-3620111 VC September 27, 2011 9:27 133

2.5 Properties of the Derivative 133

the right-circular helix σ (t)= (cos t, sin t, t) and let T (t) be its temperature at time t.

(a) What is T(t)?

(b) Find an approximate value for the temperature at t= (π/2) + 0.01.

11.

Let f (x, y, z)= (3y + 2, x2+ y2, x+ z2). Let c(t)= (cos(t), sin(t), t).

(a) Find the path p= f ◦ c and the velocity vector p(π).

(b) Find c(π ), c(π) and D f (−1, 0, π).

(c) Thinking of D f (−1, 0, π) as a linear map, find D f (−1, 0, π) (c(π)).

12.

Let h:R3→ R5and g:R2→ R3be given by h(x, y, z)= (xyz, ex z, x sin( y),−9x , 17) and g(u, v)= (v2+ 2u, π, 2

u). Find D(h◦ g)(1, 1).

13.

Suppose that a duck is swimming in the circle x = cos t, y = sin t and that the water temperature is given by the formula T = x2ey− xy3. Find dT /dt, the rate of change in temperature the duck might feel: (a) by the chain rule; (b) by expressing T in terms of t and differentiating.

14.

Let f :Rn→ Rmbe a linear mapping so that (by Exercise 28, Section 2.3) Df (x) is the matrix of f . Check the validity of the chain rule directly for linear mappings.

15.

Let f :R2→ R2; (x, y)#→ (ex+y, ex−y). Let c(t) be a path with c(0)= (0, 0) and c(0)= (1, 1). What is the tangent vector to the image of c(t) under f at t= 0?

16.

Let f (x, y)= 1/

)

x2+ y2. Compute∇ f (x, y).

17.

Write out the chain rule for each of the following functions and justify your answer in each case using Theorem 11.

(a) ∂h/∂ x, where h(x, y)= f (x, u(x, y)) (b) dh/d x, where h(x)= f (x, u(x), v(x)) (c) ∂h/∂ x, where h(x, y, z)= f (u(x, y, z),

v(x, y), w(x))

18.

Verify the chain rule for ∂h/∂ x, where h(x, y)= f (u(x, y), v(x, y)) and f (u, v)=u2+ v2

u2− v2, u(x, y)= e−x−y, v(x, y)= ex y.

19.

(a) Let y(x) be defined implicitly by G(x, y(x))= 0,

where G is a given function of two variables. Prove that if y(x) and G are differentiable, then

dy

d x = −∂G/∂ x

∂G/∂ y if ∂G

y ̸= 0.

(b) Obtain a formula analogous to that in part (a) if y1, y2are defined implicitly by

G1(x, y1(x), y2(x))= 0, G2(x, y1(x), y2(x))= 0.

(c) Let y be defined implicitly by x2+ y3+ ey= 0.

Compute dy/d x in terms of x and y.

20.

Thermodynamics texts4use the relationship

*

∂y

x

+*

z

y

+*

x

∂z

+

= −1.

Explain the meaning of this equation and prove that it is true. [HINT: Start with a relationship F(x, y, z)= 0 that defines x= f ( y, z), y = g(x, z), and z = h(x, y) and differentiate implicitly.]

21.

Dieterici’s equation of state for a gas is P(V− b)ea/RV T = RT,

where a, b, and R are constants. Regard volume V as a function of temperature T and pressure P and prove that

V

∂T =

*

R+ a

T V

+,*

RT V − ba

V2

+

.

22.

This exercise gives another example of the fact that the

chain rule is not applicable if f is not differentiable.

Consider the function

f (x, y)=

⎧ ⎨

x y2

x2+ y2 (x, y)̸= (0, 0)

0 (x, y)= (0, 0).

4See S. M. Binder, “Mathematical Methods in Elementary Thermodynamics,” J. Chem. Educ., 43 (1966): 85–92. A proper understanding of partial differentiation can be of significant use in applications; for example, see M. Feinberg, “Constitutive Equation for Ideal Gas Mixtures and Ideal Solutions as Consequences of Simple Postulates,” Chem. Eng. Sci., 32 (1977): 75–78.

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Marsden-3620111 VC September 27, 2011 9:27 134

134 Differentiation

Show that

(a) ∂f /∂ x and ∂ f /∂ y exist at (0, 0).

(b) If g(t)= (at, bt) for constants a and b, then f ◦ g is differentiable and ( f ◦ g)(0)=

ab2/(a2+ b2), but∇ f (0, 0) · g(0)= 0.

23.

Prove that if f : U ⊂ Rn → R is differentiable at x0∈ U, there is a neighborhood V of 0 ∈ Rnand a function R1: V → R such that for all h ∈ V , we have x0+ h ∈ U,

f (x0+ h) = f (x0)+ [Df (x0)]h+ R1(h) and

R1(h)

∥h∥ → 0 as h → 0.

24.

Suppose x0∈ Rnand 0≤ r1<r2. Show that there is a C1function f :Rn→ R such that f (x) = 0 for

∥x − x0∥ ≥ r2; 0 < f (x) < 1 for r1<∥x − x0∥ < r2; and f (x)= 1 for ∥x − x0∥ ≤ r1. [HINT: Apply a cubic polynomial with g(r12)= 1 and g(r22)= g(r22)= g(r12)= 0 to ∥x − x02when r1<∥x − x0∥ < r2.]

25.

Find a C1mapping f :R3→ R3that takes the vector i+ j + k emanating from the origin to i − j emanating from (1, 1, 0) and takes k emanating from (1, 1, 0) to k− i emanating from the origin.

26.

What is wrong with the following argument? Suppose w= f (x, y, z) and z = g(x, y). By the chain rule,

w

x =∂w

∂x

x

x +∂w

y

y

x +∂w

∂z

z

x =∂w

x +∂w

z

z

x. Hence, 0= (∂w/∂z)(∂z/∂x), and so ∂w/∂z = 0 or

∂z/∂ x = 0, which is, in general, absurd.

27.

Prove rules (iii) and (iv) of Theorem 10. (HINT: Use the same addition and subtraction tricks as in the

one-variable case and Theorem 8.)

28.

Show that h:Rn → Rmis differentiable if and only if each of the m components hi:Rn → R is differentiable.

(HINT: Use the coordinate projection function and the chain rule for one implication and consider

0

∥h(x) − h(x0)− Dh(x0)(x− x0)∥

∥x − x0

1

2

=

2

m

i=1[hi(x)− hi(x0)Dhi(x0)(x − x0)]2

∥x − x02 to obtain the other.)

29.

Use the chain rule and differentiation under the integral sign, namely,

d d x

3

b a

f (x, y) dy=

3

b

a

f

x(x, y) dy,

to show that d d x

3

x 0

f (x, y) dy= f (x, x) +

3

x

0

f

∂x(x, y) dy.

30.

For what integers p > 0 is

f (x)=

4

xpsin (1/x) x̸= 0

0 x= 0

differentiable? For what p is the derivative continuous?

31.

Suppose f :Rn→ R and g: Rn → Rmare differentiable. Show that the product function h(x)= f (x)g(x) from RntoRmis differentiable and that if x0and y are inRn, then [Dh(x0)]y=

f (x0){[Dg(x0)]y} + {[Df (x0)]y}g(x0).

32.

Let g(u, v)= (eu, u+ sin v) and f (x, y, z) = (xy, yz).

Compute D(g◦ f ) at (0, 1, 0) using the chain rule.

33.

Let f :R4→ R and c(t): R → R4. Suppose

∇ f (1, 1, π, e6)= (0, 1, 3, −7), c(π) = (1, 1, π, e6), and c(π)= (19, 11, 0, 1). Findd( f ◦ c)

dt when t= π.

34.

Suppose f :Rn→ Rmand g:Rp→ Rq.

(a) What must be true about the numbers n, m, p, and q for f ◦ g to make sense?

(b) What must be true about the numbers n, m, p, and q for g◦ f to make sense?

(c) When does f ◦ f make sense?

35.

If z= f (x − y), use the chain rule to show that

∂z

x +∂z

y = 0.

36.

Let w= x2+ y2+ z2, x= uv, y = u cos v, z= u sin v. Use the chain rule to findw when(u, v)= (1, 0). ∂u

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