LESSON 9
Partial Derivative and Tangent Planes
READ:
Sections 15.3, 15.4
NOTES:
The role of the derivative for functions of one variable studied back in Calculus I is taken over by the
notion of a
partial derivative
for functions of several variables. Geometrically, partial derivatives are easy
to understand.
Imagine there is a surface in 3-space with equation
z
=
f
(
x, y
), and that (
a, b, c
) is one
point on that surface, so that the point on the surface directly above (
a, b
) in the
x
,
y
-plane has
z
coordinate
c
=
f
(
a, b
). Now, take a plane parallel to the
x
,
z
-plane, passing through
b
on the
y
-axis. This plane will
intersect the surface in a curve. See figure 1B in section 15.3. On that curve the
y
coordinate never changes:
it stays constant at
b
. The
z
coordinate is computed from
z
=
f
(
x, b
) for each
x
. The
partial derivative
of
f
at the point (
a, b
) is the slope of the tangent line to that curve at the point (
a, b, c
).
To compute that partial derivative, we need only differentiate
z
=
f
(
x, b
) with respect to
x
in the good
old-fashioned Calculus I way.
For example, if
z
=
x
2
+
y
2
, then one point on this paraboloid would be
(2
,
3
,
13). The curve of intersection described above would have equation
z
=
x
2
+ 9. The derivative of
z
with respect to
x
is 2
x
.
So, putting in the
x
value of 2 at the point in question, we see the slope of the
tangent line is 4.
Another way to the say the same thing: to compute the partial derivative of
z
with respect to
x
, compute the
derivative of
f
(
x, y
) in the usual way, pretending
y
is a constant. The particular value
y
=
b
can be substituted
in after the differentiation has been carried out. That’s a bit more efficient. So, if
z
=
y
2
x
+
x
3
sin
y
, then the
partial derivative of
z
with respect to
x
at any point (
x, y
) would be given by the expression
y
2
+ 3
x
2
sin
y
,
where we have thought of
y
as a constant and
x
as the variable while computing the derivative. At the point
on the surface above the point (2
, π
) in the
x
,
y
-plane the slope of the curve of intersection described above
would be
π
2
.
The notation for the partial derivative of
z
with respect to
x
is
∂z
∂x
, so the example above would be written
as
∂z
∂x
=
y
2
+ 3
x
2
sin
y
. Other common symbols for that partial are
f
x
(
x, y
) =
y
2
+ 3
x
2
sin
y
and
∂
∂x
(
y
2
x
+
x
3
sin
y
) =
y
2
+ 3
x
2
sin
y
.
If the surface is sliced by a plane parallel to the
y
,
z
-plane, so that
x
has a fixed value along the curve of
intersection, then the slopes at various points along that curve are determined by taking the derivative of
z
=
f
(
x, y
), this time treating
x
as a constant and
y
as the variable. The result is called the partial derivative
of
z
with respect to
y
. In the example above, we would write
f
y
(
x, y
) =
∂z
∂y
= 2
yx
+
x
3
cos
y.
Geometrically, the value of the partial derivative of
z
with respect to
x
at a point tells us the rate at which
the surface is rising or falling at that point
in the
x
direction
. Imagine the positive
x
axis point east and
the positive
y
axis pointing north, in the example above. We were standing on the mountain directly above
the point in the
x
,
y