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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 ${\mathbb{R}}^2$, its partial derivatives at $(a,b)$ are the limitswhen 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 functionHolding $y$ fixed gives $f_x=2x+y$, while holding $x$ fixed gives $f_y=x+4y$. 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 areDividing either height change by $t\ne 0$ gives $3+t$ or $5+2t$; 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.

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.

A physical quantity is a property of a phenomenon, body or substance whose magnitude can be expressed by a number and a reference, usually a unit of measurement. For example, the length $L=2\mathrm{m}$ and the same length $L=200\mathrm{cm}$ have different numerical values but represent the same physical quantity. The numerical value alone is therefore incomplete.

A physical quantity must also be specified physically. A gas's pressure at a stated position and temperature, the mass of a specimen, and a detector's electric current are different measurands even if an instrument happens to display the same number for them. Dimensional analysis constrains which quantities can be added or compared, while a system of units supplies a consistent convention for expressing them.^[ref]

A system of units chooses reference units of measurement for a set of base physical quantities and specifies how other units of measurement are derived. In the International System of Units, length, mass and time are expressed using the metre, kilogram and second; a velocity has derived unit $\mathrm{m}{\mathrm{s}}^{-1}$.

A consistent system allows an equation to connect numerical values without hidden unit-conversion factors. It does not change the underlying physical relation: the same speed may be written as $1\mathrm{m}{\mathrm{s}}^{-1}$ or $3.6\mathrm{km}{\mathrm{h}}^{-1}$. The familiar formulas for electromagnetism can nevertheless contain different constants in SI and Gaussian conventions, because the definitions of the electrical quantities differ as well as their numerical scales.^[ref]