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Partial derivative Created 2026-09-24 Updated 2026-10-14

A partial derivative measures the local response of a function to one input while its other inputs stay fixed. For a function

f

on an open subset of

R^{2}

, its partial derivatives at

(a, b)

are the limits

f_{x} (a, b) = lim_{h \to 0} \frac{f ( a + h , b ) - f ( a , b )}{h}, f_{y} (a, b) = lim_{k \to 0} \frac{f ( a , b + k ) - f ( a , b )}{k},

when the respective limits exist. Each is an ordinary derivative of a one-variable slice:

x \mapsto f (x, b)

or

y \mapsto f (a, y)

. This is why the other variable is treated as a constant during partial differentiation.^[ref]

For an original example, take the polynomial function

f (x, y) = x^{2} + x y + 2 y^{2} .

Holding

y

fixed gives

f_{x} = 2 x + y

, while holding

x

fixed gives

f_{y} = x + 4 y

. At

(1, 1)

, the height is

f = 4

and the two partial derivatives are $f_{x} = 3$ and $f_{y} = 5$ . The values answer different experiments on the same mathematical model: change

x

alone, or change

y

alone.

The two exact slices through that point are

f (1 + t, 1) = 4 + 3 t + t^{2}, f (1, 1 + t) = 4 + 5 t + 2 t^{2} .

Dividing either height change by

t \neq = 0

gives

3 + t

or

5 + 2 t

; taking the limit

t \to 0

recovers the two slopes. At

t = 0.1

, the finite height changes are

0.31

and

0.52

, although their linear approximations are

0.3

and

0.5

. A derivative measures the limiting response, rather than an exact finite change.

Figure 1.
Two coordinate slices through one point give different partial derivatives
. Original contour plot of $f = x^{2} + x y + 2 y^{2}$ , with the point $(1, 1)$ and coordinate paths. The right panel compares the exact slice height changes $3 t + t^{2}$ and $5 t + 2 t^{2}$ with their tangent line predictions $3 t$ and $5 t$ . Both slices pass through height $4$ . Details.

For a function of

n

variables,

\partial_{j} f (a)

varies only its

j

th coordinate: it is the same difference quotient along

e_{j}

, the

j

th standard basis vector. This gives a practical sensitivity analysis: one can compare how a mathematical model responds to changes in its individual inputs, provided the chosen system of units and the quantities held fixed are stated. Comparing the numerical sizes of two partial derivatives in different units does not by itself identify a physically preferable input.

Existence of the partial derivatives at one point is weaker than differentiability. The example under partial derivatives do not imply continuity shows how both coordinate tests can miss a bad diagonal approach. When the first partial derivatives are continuous in a neighborhood, they do give differentiability there, allowing the total differential to combine all small input changes.

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