MAS152
Essential Mathematical Skills & Techniques
Examples 4: Partial Differentiation
1. For each of the following functions, calculate
∂f
∂2f
∂2f
∂f
and
. Show that
=
.
∂x
∂y
∂x∂y
∂y∂x
(a) x3 + 3x2 y + xy 2 + 4y 3
(b) x sin(xy)
(c) xy 2 ln(x2 + y 2 )
sin r
(d)
, where r 2 = x2 + y 2 .
r
[First show that ∂r/∂x = x/r.]
2. If f (x, y) = 3x2 y + y 3 + 4xy 2 show that x
3. If f (x, y) = cos3 (x2 − y 2 ) find
4. If V =
x
t3/2
5. If z = ln
exp
∂f
∂f
+y
= 3f.
∂x
∂y
∂f
∂f
∂f
∂f
and
. Show that y
+x
= 0.
∂x
∂y
∂x
∂y
∂V
x2
∂2V
, show that the ratio of
to
is a constant.
t
∂t
∂x2
p
x2 + y 2 , show that
∂2z ∂2z
+
= 0.
∂x2 ∂y 2
[This equation is called Laplace’s equation in two dimensions. It arises in many physical
and engineering applications.]
F ′ (r)x
∂f
=
.
6. Show that if r = x + y and f (x, y) = F (r) then
∂x
r
∂2f
x2
y2
′′
′
Show further that
=
F
(r)
+
F
(r)
and hence that
∂x2
r2
r3
2
2
∂ f
∂ f
+
= F ′′ (r) + F ′ (r)/r.
∂x2
∂y 2
2
2
2
1
Answers
∂f
∂x
∂f
(b)
∂x
∂f
(c)
∂x
∂f
(d)
∂x
∂f
= 3x2 + 2xy + 12y 2
∂y
∂f
= sin(xy) + xy cos(xy),
= x2 cos(xy)
∂y
2x2 y 2 ∂f
2xy 3
2
2
= y 2 ln(x2 + y 2) + 2
,
=
2xy
ln(x
+
y
)
+
x + y 2 ∂y
x2 + y 2
x
∂f
y
= 3 (r cos r − sin r),
= 3 (r cos r − sin r)
r
∂y
r
2
1 ∂2V
3 −5/2
x
∂V
3 −7/2
=−
= − exp
xt
+x t
4.
∂t
t
2
4 ∂x2
1. (a)
= 3x2 + 6xy + y 2,
2