From Secant Slopes to the H-Form
A secant line passes through two points on a function’s graph. If the first input is \(a\), and the second input is \(a+h\), the horizontal change between them is \(h\). The corresponding vertical change is \(f(a+h)-f(a)\). Dividing the vertical change by the horizontal change gives the secant slope.
When \(h\) is nonzero, the two inputs are distinct, so this quotient is defined whenever the function values exist. To find the instantaneous rate of change at \(a\), we examine what happens to the secant slope as the second input approaches the first. In this form, that means letting \(h\) approach zero.
The definition separates two roles. The number \(a\) is the fixed input where the derivative is being found. The variable \(h\) is an increment: it tells how far the nearby input \(a+h\) is from \(a\). The limit asks what value the quotient approaches as this increment gets arbitrarily small from either side.
What the Symbols Mean
| Symbol or expression | Role in the definition |
|---|---|
| \(a\) | The fixed input at which the derivative is evaluated |
| \(h\) | The change in input, or increment, from \(a\) |
| \(a+h\) | The nearby input reached after making the increment \(h\) |
| \(f(a+h)-f(a)\) | The change in the function’s output |
| \(h\ne 0\) | The condition that makes the quotient defined |
| \(h\to 0\) | The limiting process used to approach the instantaneous rate |
It is important to distinguish the quotient from its limit. For each allowed nonzero \(h\), the quotient gives a secant slope. The derivative is the value those slopes approach as \(h\) tends to zero. The quotient itself is not evaluated at \(h=0\); at \(h=0\), its denominator would be zero.
The limit is two-sided: \(h\) may approach zero through positive values or through negative values. A derivative at an interior input exists only if the quotient approaches the same finite value from both directions. This two-sided requirement is part of the definition, not an optional extra test.
Why the Quotient Measures a Slope
The connection to secant slopes follows directly from the slope formula. For two points with coordinates \((a,f(a))\) and \((a+h,f(a+h))\), the slope is the change in the vertical coordinate divided by the change in the horizontal coordinate. The horizontal change simplifies as follows:
Therefore, whenever \(h\ne 0\), the secant slope is exactly the difference quotient. As \(h\) approaches zero, the second point moves toward the first along the graph. If the slopes approach a finite value, that limiting value is the tangent slope and hence the instantaneous rate of change at \(a\).
Worked Example: A Quadratic at a Fixed Input
Worked Example: Find the derivative of \(f(x)=x^2\) at \(a=3\)
Start with the H-form definition. Here \(a=3\), \(f(3)=3^2=9\), and \(f(3+h)=(3+h)^2\). Thus, for \(h\ne 0\),
Expand the square and simplify the numerator:
The cancellation is valid because the quotient is considered only for \(h\ne 0\). Now take the limit as \(h\) approaches zero:
Check: The simplified quotient differs from \(6\) by exactly \(h\). For any positive tolerance \(\varepsilon\), choosing \(0<|h|<\varepsilon\) gives \(|(6+h)-6|=|h|<\varepsilon\), which verifies that the limit is \(6\). For example, \(h=0.1\) gives a secant slope of \(6.1\), while \(h=-0.1\) gives \(5.9\); both are close to \(6\), and the difference shrinks as \(|h|\) shrinks.
Worked Example: A Linear Function
Worked Example: Find the derivative of \(f(x)=4x-7\) at \(a=2\)
First calculate the function values needed in the quotient. We have \(f(2)=4(2)-7=1\), and
Substitute these expressions into the definition:
Since the quotient is \(4\) for every nonzero \(h\), its limit as \(h\) approaches zero is also \(4\):
Check: For any two distinct inputs \(2\) and \(2+h\), the change in output is \(4h\), while the change in input is \(h\). Their ratio is \(4h/h=4\). Thus the secant slopes are exactly \(4\) throughout the limiting process, agreeing with the computed derivative.
Worked Example: A Derivative That Does Not Exist
Worked Example: Test \(f(x)=|x|\) at \(a=0\)
Here \(f(0)=|0|=0\), and \(f(0+h)=|h|\). For \(h\ne 0\), the difference quotient is
If \(h>0\), then \(|h|=h\), so the quotient is \(h/h=1\). If \(h<0\), then \(|h|=-h\), so the quotient is \((-h)/h=-1\). Therefore, the values from the two sides do not approach the same number:
Check: For every positive \(h\), the quotient is \(1\); for every negative \(h\), it is \(-1\). Since these one-sided limits differ, the two-sided limit required by the derivative definition does not exist. Thus \(f'(0)\) does not exist.
A Common Mix-Up: Confusing the Input with the Increment
The notation \(a+h\) can be misread if the roles of \(a\) and \(h\) are not kept clear. The input being used for the nearby function value is \(a+h\), not \(h\) alone. The number \(h\) is only the displacement from the fixed input.
Common Mistake: Using \(f(h)\) Instead of \(f(a+h)\)
For \(f(x)=x^2\) at \(a=3\), a mistaken setup might be
This uses \(h\) as the nearby input, even though the nearby input should be \(3+h\). It therefore compares the point \(h\) with the point \(3\), rather than comparing the point \(3+h\) with the point \(3\). The correct setup is
Taking the limit gives \(6\), as shown above. A useful check is to identify the two inputs explicitly: they are \(a\) and \(a+h\), so their difference must be \(h\).
Conditions and Interpretation
To use the H-form at an interior point \(a\), the function must be defined at \(a\) and at inputs sufficiently close to \(a\). The quotient must be defined for nonzero increments near zero, and its two-sided limit must exist as a finite number. If the limit fails to exist, the derivative at that point is not defined by this definition.
The derivative is a local measurement. It describes the function’s output change per unit of input change near one particular input. If the input represents time and the output represents distance, the quotient is an average rate of change over a short time interval, and the limit is the instantaneous rate at the chosen time. The units of the quotient are output units divided by input units.
The increment \(h\) is allowed to be positive or negative. A positive \(h\) uses an input to the right of \(a\) on a number line; a negative \(h\) uses one to the left. Both directions matter for the ordinary two-sided derivative at an interior point. The limiting process does not require selecting one fixed small increment: it asks what happens for increments arbitrarily close to zero.
Check Your Understanding
Use the definition and the roles of \(a\) and \(h\) to answer these questions.
- In the H-form definition, which quantity is fixed, and which quantity approaches zero?
- For \(f(x)=x^2\) at \(a=2\), write the difference quotient before simplifying it.
- Why must \(h\) be nonzero in the quotient even though the limit is taken as \(h\) approaches zero?
- For \(f(x)=|x|\) at \(a=0\), what are the limits of the quotient from positive and negative values of \(h\)?
- In the secant-slope interpretation, what two inputs are being compared?
