Differentials and the Total Differential

From Input Changes to a Linear Prediction

A function of two variables can change when either input changes, or when both change together. At a fixed point, its partial derivatives describe the rates of change in the two coordinate directions. The total differential combines those two rates into one linear expression for the first-order part of the function’s change.

The inputs’ changes will be denoted by \(dx\) and \(dy\). In this setting, these symbols mean actual increments from a chosen base point. The differential \(df\) is then defined using those increments and the partial derivatives at that point. It is important to keep the meaning of \(df\) precise: for a general differentiable function, it is the linear part of the function change, not necessarily the exact change itself.

Input Increments and Function Change

Let \(f\) be a real-valued function of two variables, and choose a base point \((a,b)\) in its domain. If the input moves to \((a+dx,b+dy)\), then \(dx\) and \(dy\) are the changes in the first and second coordinates:

The actual change in the function value is the new value minus the old value. It is often written \(\Delta f\):

This definition gives the exact change, whenever both input points are in the domain. To define the total differential, we also require \(f\) to be differentiable at \((a,b)\). Differentiability means that near this point the function’s change consists of a linear part plus a remainder that becomes negligible relative to the size of the input change.

Here \(f_x(a,b)\) and \(f_y(a,b)\) are the partial derivatives evaluated at the base point, not at the new point. In the common notation \(z=f(x,y)\), one may write \(dz\) in place of \(df\):

In this version, the formula is understood at a particular base point \((x,y)\), with the partial derivatives evaluated there. The symbols \(dx\) and \(dy\) specify the input increments being considered.

Why the Formula Works

The formula is a direct expression of differentiability. At \((a,b)\), differentiability says that the function can be written as a linear approximation plus a remainder:

where the remainder \(r\) satisfies

for nonzero increments. Subtracting \(f(a,b)\) from both sides yields the actual change:

The first two terms are exactly \(df\), so \(\Delta f=df+r(dx,dy)\). This establishes both the formula and its role: \(df\) is the linear part of the change, while the remainder accounts for what that linear part does not capture.

The expression is linear in the increments: doubling both increments doubles \(df\), and adding two pairs of increments adds their predicted differentials. That linearity is what makes \(df\) the derivative’s contribution to the function change.

Worked Example: A Quadratic Function with a Linear Term

Worked Example: A Product and a Mixed Increment

Worked Example: A Function with Two Squared Inputs

A Common Mistake: Treating the Differential as an Exact Change

A frequent error is to write \(\Delta f=df\) for every input change. The product example shows why that is not generally valid. For \(f(x,y)=xy\) at \((1,2)\) with \(dx=\frac{1}{4}\) and \(dy=-\frac{1}{2}\), the incorrect calculation would be:

But direct substitution gives \(\Delta f=-\frac{1}{8}\), not \(0\). The corrected relationship is

where the remainder is not necessarily zero. In this product example, the remainder is \(dx\,dy=-\frac{1}{8}\). The differential and the actual change agree in special cases, such as when the function is linear, but not for every differentiable function or every finite input increment.

Why the Total Differential Matters

The total differential gives a compact way to express how the two coordinate directions contribute to the linear part of a function’s change. Each partial derivative supplies a coefficient: the \(x\)-direction contribution is \(f_x(a,b)\,dx\), and the \(y\)-direction contribution is \(f_y(a,b)\,dy\). Adding them gives the derivative’s action on the pair of input increments.

The distinction between a linear prediction and an exact change matters whenever \(dx\) and \(dy\) are not zero. The definition of differentiability controls the remainder as the increments approach zero; it does not assert that the remainder vanishes for each chosen pair. Keeping \(\Delta f\) and \(df\) separate makes the meaning of the formula clear and prevents an approximation from being mistaken for an identity.