TA. The Tangent Approximation
1.
Partial derivatives
Let w
=
f
(x, y) be a function of two variables. Its graph is a surface in xyzspace, as
pictured.
w
Fix a value y
=
yo and just let x vary. You get a function of
one
variable,
(1) w
=
f
(x, yo),
the
partial function
for y
=
yo.
Its graph is a curve in the vertical plane y
=
yo, whose slope at the
point
P where x
=
xo is given by the derivative
d
(2)
f
dx
x
Y
O
,
Or
afl
.
x
xo
(XO,YO)
We call (2) the
partial derivative
of
f
with respect to x at the point (xo, yo); the right side
of (2) is the standard notation for it. The partial derivative is just the ordinary derivative
of the partial function

it is calculated by holding one variable fixed and differentiating
with respect to the other variable. Other notations for this partial derivative are
the first is convenient for including the specific point; the second is common in science
and engineering, where you are just dealing with relations between variables and don't
mention the function explicitly; the third and fourth indicate the point by just using a
single subscript.
Analogously, fixing x
=
xo and letting y vary, we get the partial function w
=
f
(xo, y),
whose graph lies in the vertical plane x
=
xo, and whose slope at
P is the
partial derivative
off with respect to
y; the notations are
The partial derivatives d
f
ldx and d
f
ldy depend on (xo, yo) and are therefore functions of
x andy.
Written as aw/dx, the partial derivatrive gives the rate of change of w with respect to
x alone, at the point (xo, YO): it tells how fast w is increasing as x increases, when y is held
constant.
For a function of three or more variables, w
=
f
(x, y, z,
. .
.),
we cannot draw graphs any
more, but the idea behind partial differentiation remains the same: to define the partial
derivative with respect to x, for instance, hold all the other variables constant and take the
ordinary derivative with respect to x; the notations are the same as above: