Difference Quotient and the Definition of the Derivative
The derivative of f at a is the limit of the difference quotient f′(a) = limh→0 [f(a + h) − f(a)]/h, which gives the instantaneous rate of change of f and the slope of the tangent line at the point (a, f(a)). In brief: for f(x) = x2, the difference quotient at x is (2xh + h2)/h = 2x + h, which tends to 2x as h → 0, so f′(x) = 2x. At x = 3 the slope of the tangent is 6.
The difference quotient measures the average rate of change over a small interval, and the derivative is what remains when the interval shrinks to a point. This single idea underlies velocity, acceleration, marginal cost, growth rates and the slope of a curve. This guide explains the difference quotient, the definition of the derivative, how to compute derivatives from the definition for several functions, the relation to continuity, notation and one-sided derivatives, with a table of values approaching the limit.
What is the difference quotient?
For a function f and two inputs a and a + h, the difference quotient is [f(a + h) − f(a)]/h. It is the average rate of change of f on [a, a + h], and the slope of the secant line through (a, f(a)) and (a + h, f(a + h)). For a distance function s(t), it is the average speed over a time interval of length h.
Example: for s(t) = 5t2 meters, the average speed between t = 2 and t = 3 is (45 − 20)/1 = 25 m/s. Between t = 2 and t = 2.1: (22.05 − 20)/0.1 = 20.5 m/s. Between t = 2 and 2.01: (20.2005 − 20)/0.01 = 20.05 m/s. The averages approach 20 m/s, the instantaneous speed at t = 2.
What is the definition of the derivative?
The derivative of f at a is
f′(a) = limh→0 [f(a + h) − f(a)]/h, or equivalently, f′(a) = limx→a [f(x) − f(a)]/(x − a),
if the limit exists. If it exists, f is differentiable at a. The derivative function f′(x) assigns to each x the number f′(x). Geometrically, f′(a) is the slope of the tangent line at (a, f(a)): the limit of the secant lines as the second point approaches the first.
How do you compute derivatives from the definition?
- Write f(x + h) and f(x).
- Form [f(x + h) − f(x)]/h.
- Simplify algebraically so that h cancels in the denominator.
- Take the limit h → 0.
Example 1: f(x) = x2
[(x + h)2 − x2]/h = (2xh + h2)/h = 2x + h → 2x. So (x2)′ = 2x. At x = 3 the slope is 6, and the tangent line is y − 9 = 6(x − 3), i.e. y = 6x − 9.
Example 2: f(x) = x3
[(x + h)3 − x3]/h = (3x2h + 3xh2 + h3)/h = 3x2 + 3xh + h2 → 3x2.
Example 3: f(x) = 1/x
[1/(x + h) − 1/x]/h = [(x − x − h)/(x(x + h))]/h = −1/(x(x + h)) → −1/x2. At x = 2 the slope is −1/4.
Example 4: f(x) = √x
[√(x + h) − √x]/h = [(x + h) − x]/(h(√(x + h) + √x)) = 1/(√(x + h) + √x) → 1/(2√x), for x > 0.
Example 5: a linear function and a constant
For f(x) = mx + b: [m(x + h) + b − mx − b]/h = m, so f′(x) = m. For a constant f(x) = c: the difference is 0, so f′(x) = 0.
Example 6: f(x) = |x| at 0
[|h| − 0]/h = |h|/h, which is 1 for h > 0 and −1 for h < 0. The one-sided limits differ, so |x| is not differentiable at 0 (a corner), but f′(x) = 1 for x > 0 and −1 for x < 0.
Example 7: a physical meaning
A ball thrown upward has the height h(t) = 20t − 5t2 meters. [h(t + k) − h(t)]/k = 20 − 10t − 5k → 20 − 10t m/s. At t = 1: 10 m/s (rising); at t = 2: 0 (the top); at t = 3: −10 m/s (falling).
What is the relation between differentiability and continuity?
If f is differentiable at a, then f is continuous at a. Proof: f(a + h) − f(a) = h · [f(a + h) − f(a)]/h → 0 · f′(a) = 0. The converse is false: |x| is continuous at 0 but not differentiable. A function fails to be differentiable at a corner (|x| at 0), at a vertical tangent (∛x at 0, where the slope is infinite) and at a discontinuity.
The one-sided derivatives f′+(a) and f′−(a) use h → 0+ and h → 0−. The derivative exists if and only if both exist and are equal.
Which notations are used?
- Lagrange: f′(x), f″(x) for the second derivative.
- Leibniz: dy/dx, d2y/dx2, and d/dx for the operator. It suggests a quotient of small changes Δy/Δx.
- Newton: a dot over the variable, ẋ, for derivatives with respect to time (physics).
- Euler: Df or Dxf.
Table of difference quotients
| h | f(3 + h) | f(3 + h) − f(3) | Quotient (6 + h) |
|---|---|---|---|
| 1 | 16 | 7 | 7 |
| 0.1 | 9.61 | 0.61 | 6.1 |
| 0.01 | 9.0601 | 0.0601 | 6.01 |
| 0.001 | 9.006001 | 0.006001 | 6.001 |
| −0.001 | 8.994001 | −0.005999 | 5.999 |
From both sides the quotient approaches 6, so f′(3) = 6.
Where does the derivative come from?
The problem of the tangent was studied by Fermat and Descartes in the 1630s, and the velocity problem by Galileo and others. Isaac Newton (1665–1666) and Gottfried Wilhelm Leibniz (1675–1684), independently, created the calculus with the derivative as the central notion; a bitter priority dispute followed. The rigorous definition with limits was given by Cauchy in the 1820s, replacing the “infinitely small” quantities.
Where is the derivative used?
Velocity is the derivative of position and acceleration is the derivative of velocity. In economics, marginal cost is the derivative of the cost function. In biology, the growth rate of a population is the derivative of its size. The derivative is used to find the tangent line (a linear approximation: f(x) ≈ f(a) + f′(a)(x − a)), and to find maxima and minima.
Which mistakes are common?
Substituting h = 0 too early
The quotient has the form 0/0. Simplify first, cancelling h, and only then take the limit.
Forgetting the f(x) term
The numerator is f(x + h) − f(x), not just f(x + h).
Confusing f′(a) with the derivative of the constant f(a)
f′(a) is a number (the slope at a). The derivative of the constant f(a) is 0.
Assuming that continuous means differentiable
|x| and ∛x are continuous at 0 but not differentiable there.