For a one-variable curve, the
tangent line
is the best straight-line approximation near a point. A surface
z = f(x, y) has two
partial derivatives,
one slope in the x direction and one in the
y direction, and together they pin down the best flat approximation
— a tangent plane. At a point (a, b) it is
z = f(a, b) + f_x(a, b)\,(x - a) + f_y(a, b)\,(y - b).
Read it as a recipe: start at the height f(a, b), then add the rise
from moving (x - a) in the x direction at
slope f_x, and the rise from moving (y - b)
in the y direction at slope f_y. Each
x-slice of this plane is exactly the tangent line to the
corresponding slice of the surface.
Building the plane, step by step
Build the tangent plane to the paraboloid f(x, y) = x^2 + y^2 at the
point (a, b) = (1, 1), then use it to estimate
f(1.1, 0.9) without squaring anything.
Step 1 — the partials. Differentiate in each variable, holding the other
constant:
f_x = 2x, \qquad f_y = 2y.
Step 2 — evaluate at the point (1, 1). The height
and the two slopes are
f(1, 1) = 1 + 1 = 2, \qquad f_x(1, 1) = 2, \qquad f_y(1, 1) = 2.
Step 3 — assemble the plane. Substitute into the template:
z = 2 + 2\,(x - 1) + 2\,(y - 1).
Step 4 — read it as a linearization. The right-hand side is the
linear approximation L(x, y) of
f near (1, 1):
L(x, y) = 2 + 2(x - 1) + 2(y - 1).
Step 5 — estimate f(1.1, 0.9). Here
x - 1 = 0.1 and y - 1 = -0.1, so the two
corrections cancel:
L(1.1, 0.9) = 2 + 2(0.1) + 2(-0.1) = 2 + 0.2 - 0.2 = 2.
Step 6 — check against the truth. The exact value is
f(1.1, 0.9) = 1.1^2 + 0.9^2 = 1.21 + 0.81 = 2.02.
The estimate 2 is off by only 0.02 — the
error is second-order in the small step, exactly as the tangent-line story promised, now in
two variables.
The total differential
Writing dx = x - a and dy = y - b for the
small displacements, the change in height predicted by the plane is the
total differential
df = f_x\, dx + f_y\, dy.
It is the workhorse of error propagation: a small wiggle dx in one
input and dy in the other produce, to first order, a change
df in the output. Drag the point a below
to watch the tangent line track the slice of the paraboloid and read off the linearization
there.
Let f be differentiable at (a, b).
Then:
-
The tangent plane to z = f(x, y) at
(a, b) is
z = f(a, b) + f_x(a, b)(x - a) + f_y(a, b)(y - b).
-
The linear approximation
L(x, y) equals the right-hand side, and
f(x, y) \approx L(x, y) near (a, b),
with error vanishing faster than the distance to (a, b).
-
The total differential
df = f_x\,dx + f_y\,dy gives the first-order change in
f from displacements dx, dy.
In one variable, "the derivative exists" and "the function is differentiable" are the same
statement. In two variables they come apart, and the gap is a genuine trap. A function can
have both partial derivatives at a point and yet not be
differentiable there — not even continuous. The standard specimen is
f(x, y) = \frac{xy}{x^2 + y^2} \quad (f(0, 0) = 0).
Along the axes f is identically zero, so
f_x(0, 0) = f_y(0, 0) = 0 both exist. But we already saw this
function has
no limit at the origin
— so it is not even continuous there, let alone differentiable. The partials see only two
directions; differentiability demands the tangent plane approximate f
from every direction at once.
The clean sufficient condition: if f_x and
f_y exist and are continuous near
(a, b), then f is differentiable there
(such an f is called C^1). For
everyday functions — polynomials, exponentials, sines — this always holds, which is why the
tangent-plane formula can be applied without anxiety.