Quadratic Derivatives by Factoring

Why Factoring Helps

A derivative measures the limiting rate of change at an input. From first principles, that rate is defined using a limit of secant slopes. For a quadratic polynomial, direct substitution into the difference quotient initially produces a fraction with numerator and denominator both equal to zero. Factoring the numerator reveals why the quotient nevertheless has a finite limit.

The key is to keep two ideas separate: the increment \(h\) is nonzero in the difference quotient, but it approaches zero when we take the limit. We simplify the quotient for \(h\ne 0\) first; only then do we evaluate its limit as \(h\) approaches zero.

The numerator compares the function value at \(x_0+h\) with its value at \(x_0\). Dividing by \(h\) gives the average rate of change over that input change. If the quotient has a finite limit, that limit is the derivative at \(x_0\).

A Specified Input: Expand, Subtract, Factor

Why the Increment Factor Cancels

In the example, the numerator became \(7h+h^2\), which factors as \(h(7+h)\). The denominator is also \(h\), so the factor can be cancelled provided \(h\ne 0\). That condition is part of the definition: the quotient is considered for nonzero increments and then its limit is taken as the increments approach zero.

This works because a limit depends on the values of an expression arbitrarily near the point, not on whether the expression is defined at the point itself. In this case, after cancellation the quotient is a simple linear expression in \(h\), whose limit can be read off directly.

Quadratics at Arbitrary Inputs

The same calculation works at any real input. Let \(f(x)=Ax^2+Bx+C\), where \(A\ne 0\), and let \(x_0\) be any real number. The polynomial is defined at \(x_0\) and at all nearby inputs. Expand the value at \(x_0+h\), subtract the value at \(x_0\), and factor the increment.

Carrying out each step gives the complete derivation:

As \(h\to 0\), the term \(Ah\) approaches zero. Therefore,

This proves the result for every real \(x_0\), not just for one chosen example. The constant term \(C\) disappears because the same \(C\) occurs in both function values and cancels in their difference.

Applying the Result Carefully

A Common Mistake

A tempting but incorrect calculation is to substitute \(h=0\) into the original difference quotient and conclude that the derivative is undefined:

This only shows that the quotient itself is not defined at \(h=0\). The derivative is a limit of the quotient as nonzero \(h\) approaches zero; it is not the quotient evaluated at zero.

For example, for \(q(x)=2x^2-3x+4\), the quotient is \(\frac{h(4x+2h-3)}{h}\) when \(h\ne 0\). Cancel the common factor under that condition, then take the limit of \(4x+2h-3\). The limit is \(4x-3\). The cancellation does not make the original fraction defined at zero; it identifies a simpler expression with the same values for all nonzero increments.

Why This Method Matters

Factoring makes the defining limit manageable by exposing the part of the change that is proportional to the increment. Once that increment factor is cancelled, the remaining expression describes how the average rate of change behaves as the interval shrinks. Its limit gives the instantaneous rate of change, or equivalently the tangent slope at the input.

The method also makes clear what is special about quadratics: after expansion and subtraction, every surviving numerator term contains \(h\). Cancelling that factor leaves a term that approaches a fixed value as \(h\to 0\). For a quadratic, the calculation works at every real input, as the arbitrary-input derivation shows.