The Increment Form of the Difference Quotient

From a Nearby Input to an Increment

The derivative at \(x=a\) describes the limiting slope of secant lines whose input values approach \(a\). In the difference quotient, the nearby input is often written as \(x\). There is another useful way to describe the same nearby input: specify how far it is from \(a\). Call that signed distance \(h\). Then the nearby input is \(a+h\).

This change of variable gives the increment form of the derivative definition. It does not introduce a different derivative. It rewrites the same limit so that the displacement from the fixed point is explicit.

The numerator compares the function values at the nearby input \(a+h\) and the fixed input \(a\). The denominator \(h\) is the change in input. Thus, for each allowed nonzero \(h\), the quotient is the average rate of change over those two inputs, or equivalently the slope of the secant line through their graph points.

Why the Increment Must Be Nonzero

The difference quotient compares two distinct input values. If \(h=0\), then \(a+h=a\): the two inputs coincide, and the denominator is zero. The quotient is therefore not defined at \(h=0\). This is not a problem for the limit, because a limit as \(h\) approaches zero concerns values of \(h\) near zero, not the value \(h=0\) itself.

That restriction is essential even when the numerator also becomes zero at \(h=0\). A fraction with numerator zero and denominator zero is still undefined; it cannot be evaluated by substituting \(h=0\). The limit asks whether the quotient values for nonzero increments approach a number as the increments get arbitrarily close to zero.

Because \(h\) is a signed increment, it can be positive or negative. If \(h>0\), then \(a+h\) lies to the right of \(a\). If \(h<0\), it lies to the left. When \(a\) is an interior point of the domain, sufficiently small increments of both signs are allowed.

Why the Two Limit Forms Agree

The usual difference quotient uses \(x\), the nearby input. The increment form uses \(h\), the displacement from the fixed input. The relationship between them is exact:

In particular, the distance from the input to \(a\) is the size of the increment:

Thus \(x\) approaches \(a\) precisely when \(h\) approaches zero. Also, \(x\ne a\) precisely when \(h\ne0\). These facts ensure that the two versions of the difference quotient have the same values, expressed with different variables.

For completeness, the equivalence also follows directly from the definition of a limit. Suppose the quotient in the \(x\)-form approaches a number \(L\) as \(x\to a\), with \(x\ne a\). Given any tolerance \(\varepsilon>0\), the limit definition supplies a distance \(\delta>0\) such that the quotient is within \(\varepsilon\) of \(L\) whenever \(0<|x-a|<\delta\), for inputs in the domain. Set \(x=a+h\). Then \(0<|h|<\delta\) is exactly the same condition, and the two quotients are equal. Therefore the quotient in the \(h\)-form is within \(\varepsilon\) of \(L\) whenever \(0<|h|<\delta\). The converse follows by setting \(h=x-a\) in the same reasoning. The limits, when they exist, are equal.

Worked Example: Tracking Both Signs of the Increment

Domain Restrictions on the Increment

The expression \(a+h\) must be an input where \(f\) is defined. Consequently, the increment is not always free to take every real value. If \(a\) is an interior point of the domain, there is some open interval around \(a\) on which \(f\) is defined. In that case, all sufficiently small positive and negative increments are permitted, except zero in the quotient.

In the derivative limit, increments far from zero do not determine the limit. What matters is that every sufficiently small allowed nonzero increment is considered. For a point at the edge of a domain, only one sign of increment may be available. The usual two-sided derivative definition at an interior point assumes an open interval around \(a\); a one-sided limit requires separate attention to the side from which the increment approaches zero.

A Common Mistake: Using the Wrong Denominator

When replacing \(x\) with \(a+h\), the denominator \(x-a\) becomes \(h\), not \(a+h\). The denominator measures the change from the fixed input \(a\), rather than the nearby input itself.

For a concrete check, let \(f(x)=x^2\) and take \(h=1\). The nearby input is \(x=2+1=3\), so the secant quotient is

The incorrect denominator \(2+h=3\) would instead give \((9-4)/3=5/3\), which is not the secant quotient between inputs \(2\) and \(3\). Using the increment form gives the correction directly:

The denominator must agree with the actual input change. Here that change is \(3-2=1=h\), which checks the corrected calculation.

Using the Increment Form Carefully

When rewriting a derivative definition at \(a\), keep three details visible:

• Substitute \(a+h\) for the nearby input \(x\).
• Replace the input difference \(x-a\) with \(h\), and keep \(h\ne0\) in the quotient.
• Require \(a+h\) to lie in the domain of the function.

Then interpret \(h\to0\) as the nearby input \(a+h\) approaching \(a\). At an interior point, increments may approach zero from either side. The increment notation makes the changing input especially clear, while preserving the same difference quotient and the same derivative limit.