Deriving The Tangent Slope Of A Quadratic

From Nearby Secants to a Tangent Slope

A tangent slope describes the direction of a curve at one particular point. For the parabola \(f(x)=x^2\), we want to find that slope at any input, not just estimate it from a graph.

The earlier material on secant slopes gives a useful way to approach the question. Choose an input \(a\), then compare the point on the curve at \(a\) with a nearby point at \(a+h\). The slope of the line through those two points is a secant slope. As the increment \(h\) approaches zero, the second point approaches the first. If the secant slopes approach a finite value, that limit is the tangent slope at \(a\).

The key is to simplify the secant slope while \(h\) is still nonzero. For a quadratic, the algebra reduces the quotient to an expression whose limit is straightforward.

Set Up the Difference Quotient

Fix an arbitrary real input \(a\). The corresponding point on the graph is \((a,f(a))\). Move horizontally by an increment \(h\), where \(h\ne 0\); the nearby input is \(a+h\), and its point on the graph is \((a+h,f(a+h))\).

The slope of the secant line through these points is change in output divided by change in input:

This is the difference quotient with an increment. The condition \(h\ne 0\) matters: if \(h=0\), the two inputs are identical, so there are not two distinct points defining a secant line, and the denominator of the quotient would be zero.

For \(f(x)=x^2\), the output at the first input is \(f(a)=a^2\), and the output at the nearby input is \(f(a+h)=(a+h)^2\). Thus the secant slope is

The tangent slope at \(a\) is the limit of these secant slopes as \(h\) approaches zero, provided that this limit exists:

Here \(m_{\mathrm{tan}}(a)\) denotes the tangent slope at the point with input \(a\). The quotient describes the secant slopes for nonzero \(h\); the limit describes their behavior as the second point approaches the first.

Deriving the Slope at an Arbitrary Input

Expand the square in the numerator, combine like terms, and factor out \(h\). Since the quotient is being considered only for \(h\ne 0\), the common factor can then be cancelled:

This expression gives the same secant slope as the original quotient whenever \(h\ne 0\). Now take its limit as \(h\) approaches zero:

The calculation works for every real \(a\). The choice of \(a\) was arbitrary, and the limit exists because \(2a+h\) approaches \(2a\) as \(h\) approaches zero from either side. In particular, no restriction to positive increments is needed: both left-hand and right-hand secants approach the same slope.

Why Simplifying Before Taking the Limit Works

The original quotient is not defined at \(h=0\), because its denominator is zero there. But finding the limit does not require evaluating that quotient at \(h=0\). A limit asks what values the expression approaches when \(h\) is close to zero, with \(h\ne 0\).

For every nonzero \(h\), cancelling the common factor \(h\) gives \(2a+h\). Thus, throughout the punctured neighborhood around zero, the original secant-slope expression has exactly the same values as \(2a+h\). Since \(2a+h\) approaches \(2a\), the secant slopes approach \(2a\) as well. This establishes the claimed tangent slope directly from the limit definition.

Worked Example: Tangent Slope at the Origin

Worked Example: Tangent Slope at a Positive Input

Worked Example: Tangent Slope at a Negative Input

A Common Mistake: Substituting Zero Too Soon

A tempting but invalid approach is to set \(h=0\) in the original difference quotient. For an arbitrary \(a\), that gives

This is not a tangent slope; it is an undefined quotient. The error is treating the limit as though it were ordinary substitution into the unsimplified expression. The correction is to keep \(h\ne 0\), simplify first, and then take the limit:

Cancelling \(h\) is valid only because the quotient is being considered for nonzero \(h\). The simplified expression makes the limiting behavior visible without claiming that the original quotient is defined at zero.

What the Result Tells Us

The formula \(2a\) describes how the tangent slope varies across the parabola. At the origin it is zero; at positive inputs it is positive; at negative inputs it is negative. The values grow in magnitude as the input moves farther from zero. These conclusions follow from the formula for the tangent slope, rather than from estimating individual graph segments.

The distinction between a secant slope and a tangent slope remains essential. A secant slope uses two distinct inputs and an increment \(h\ne 0\). The tangent slope is the limit of those secant slopes as the increment approaches zero. In this case, algebra gives a simple expression for every nearby secant slope, and the limit exists at every real input.

The method is useful because it turns the geometric idea of a tangent into a precise calculation. It also shows why the answer depends on the input: the secant slopes simplify to \(2a+h\), so the part that remains when the increment vanishes is determined by \(a\). The result is not a single slope for the whole parabola; each point has its own tangent slope.