Increments and First-Order Change

From Input Changes to Output Change

A function of several variables assigns one output to a collection of inputs. If the inputs are changed, the output generally changes too. Keeping track of these changes gives a natural way to use partial derivatives together: each partial derivative describes the rate of change in one coordinate direction, while the corresponding input increment records how far that coordinate moves.

The resulting expression is a candidate first-order change. It combines the coordinatewise rates at a base point with the input increments. The word “candidate” matters: having partial derivatives at the base point gives us this expression, but by itself does not guarantee that it accurately approximates the actual change when several inputs move together.

Input and Output Increments

Let \(f:A\to\mathbb{R}\) be a scalar-valued function, where \(A\) is a subset of \(\mathbb{R}^n\). Choose a base point \(\mathbf{a}=(a_1,\ldots,a_n)\) in the domain and an input increment \(\mathbf{h}=(h_1,\ldots,h_n)\) such that \(\mathbf{a}+\mathbf{h}\) is also in the domain. The new input is \(\mathbf{a}+\mathbf{h}\); the change in the input is \(\mathbf{h}\).

For two inputs \(x\) and \(y\), we usually write the increments as \(\Delta x\) and \(\Delta y\). If the base point is \((a,b)\), the new point is \((a+\Delta x,b+\Delta y)\), and the output increment is

The output increment is an exact difference. It is not a derivative and is not automatically equal to a formula built from partial derivatives.

Constructing the Candidate First-Order Change

Suppose the first partial derivatives of \(f\) exist at the base point \(\mathbf{a}\). For a small change in coordinate \(i\) alone, the partial derivative \(\frac{\partial f}{\partial x_i}(\mathbf{a})\) gives the coordinatewise rate at that point. Multiplying this rate by the corresponding input increment gives that coordinate’s first-order contribution.

This expression is linear in the increments: doubling every input increment doubles the candidate, and adding two increment vectors adds their candidate changes. For two variables, the first term uses the change in \(x\) and the \(x\)-partial; the second term uses the change in \(y\) and the \(y\)-partial. Matching each derivative with its own coordinate increment is essential.

Why These Coefficients Appear

The formula is motivated by the meaning of a partial derivative. Along coordinate \(i\), with all other coordinates fixed at their base-point values, the derivative is the limit of the output change divided by the coordinate increment:

where \(\mathbf{e}_i\) is the vector whose \(i\)-th coordinate is \(1\) and whose other coordinates are \(0\). Thus the partial derivative supplies the first-order rate in that coordinate direction.

Here is a useful justification for combining those rates. If a first-order rule is linear in the input increment, then its value on a general increment is determined by its values on the coordinate directions. Indeed, \(\mathbf{h}=\sum_{i=1}^{n}h_i\mathbf{e}_i\). By linearity, the rule’s value at \(\mathbf{h}\) must be the sum of \(h_i\) times its value at \(\mathbf{e}_i\). The coordinatewise rates are the partial derivatives, so this gives the stated candidate formula.

This argument explains why the partial derivatives are the coefficients of a linear first-order candidate. It does not establish that the candidate matches the actual output change for every function with partial derivatives at \(\mathbf{a}\). The actual change involves moving all coordinates, and coordinatewise rates at the base point alone do not settle how well the combined change is captured. The later study of differentiability addresses the precise conditions for a valid first-order approximation.

Worked Examples

A Common Sign Error

In the product example, a common mistake is to use the correct partial derivatives but forget that \(\Delta y=-0.2\). That incorrect calculation would be

The second term must use the signed increment, not its absolute value. The corrected calculation is

A quick check is to write each derivative and increment as a matched pair before evaluating: \(f_x\Delta x\) and \(f_y\Delta y\). This makes it harder to lose the sign or attach a derivative to the wrong coordinate.

Interpreting the Candidate Carefully

The output increment \(\Delta f\) is always the difference between two function values. The candidate \(L_{\mathbf{a}}(\mathbf{h})\) is formed from the partial derivatives at the base point and the input increments. They are different quantities, even when their numerical values are close.

For a single coordinate change, the definition of the corresponding partial derivative explains the coordinatewise rate. For simultaneous changes, the candidate adds the contributions suggested by those rates. Whether that sum accurately captures the actual output change is a further question: partial derivatives at one point alone are not a guarantee.

These distinctions are useful whenever a problem asks how a function changes near a chosen input. Begin by identifying the base point and the signed input increments. Then compute the partial derivatives at that base point, form the candidate, and keep it distinct from the exact output increment unless the required approximation conditions have been established.