Increments and the definition of the derivative

Leyla is travelling by intercity bus. The bus covers 240 km in 3 hours, so its average speed is 80 km/h. Yet the speedometer never showed a steady 80: 0 at a traffic light, 90 on a straight road, 50 on a bend. Every moment has its own speed. The question “what is the speed at one particular moment?” led the mathematicians of the 17th century to the idea of the derivative. In this lesson we follow the same path step by step: first we learn to measure change (increments), then the average rate of change, and then what happens when the interval shrinks towards zero.

The increment of the argument and of the function

The derivative describes change, so first we learn to measure change. Let y = f(x) be a function and x₀ a point of its domain. The argument moves from x₀ to another value x, and the value of the function moves from f(x₀) to f(x). We write these changes with the Greek letter Δ (“delta”), which here means “difference”.

Picture it on the graph: you stand at the point A(x₀, f(x₀)) of the graph. First you take a horizontal step of Δx, then you climb vertically by Δf until you are back on the graph (if Δf < 0, you go down). This brings you to a second point of the graph, B(x₀ + Δx, f(x₀ + Δx)). Δx is the width of this “stair step” and Δf is its height.

The average rate of change and the secant

An increment alone says little: you can cover 10 km in 5 minutes or in 5 hours. To know how fast something changes, we divide the increment of the function by the increment of the argument. The ratio Δf / Δx is the average rate of change: by how many units the function changes, on average, when the argument changes by 1 unit. Its unit is “unit of the function per unit of the argument”: km/h, cm per year, manat per month.

Now the geometry. The straight line through the points A(x₀, f(x₀)) and B(x₀ + Δx, f(x₀ + Δx)) of the graph is called a secant (it “cuts” the graph in at least two points). From linear functions you know that the slope of a line through two points is “vertical difference over horizontal difference”. For A and B the vertical difference is Δf and the horizontal one is Δx. So the average rate of change is the slope of the secant: algebra and geometry say the same thing.

Δx → 0: from the secant to the tangent

An average speed describes a whole interval, but we want the speed at one moment. A natural idea: make the interval shorter and shorter. Let us compute for f(x) = x² and x₀ = 1. By the general formula above, Δf = 2x₀Δx + (Δx)² = 2Δx + (Δx)², so Δf / Δx = 2 + Δx. Now let Δx approach zero from the right and from the left:

The ratio gets as close to 2 as we like: the smaller Δx is, the smaller the gap between Δf / Δx and 2 (the gap is exactly Δx). Mathematicians write this as lim (Δx→0) Δf / Δx = 2 and read it: “the limit of Δf / Δx as Δx tends to zero is 2”. Note: Δx never becomes zero. With Δx = 0 we would get the meaningless expression 0 / 0. A limit answers the question “which number does the ratio approach?”, not “what is the ratio when Δx = 0?”.

The geometric picture is the same. As Δx shrinks, the point B slides along the graph towards A and the secant turns around A. In the limit it reaches a limiting position: the line tangent to the graph at A. The slope of the tangent is the limit of the slopes of the secants; in our example it is 2. The same holds for the bus: the average speeds over shorter and shorter time intervals approach the instantaneous speed that the speedometer shows.

The definition of the derivative

The derivative has several notations: f′(x) (“f prime of x”), y′ (“y prime”) and dy/dx (“dee y by dee x”). The last one recalls the definition: a tiny Δy divided by a tiny Δx. An important idea: if we let the point x₀ vary, we get a number f′(x) for every x. So the derivative is a new function. For example, for f(x) = x² we have f′(x) = 2x (we prove it below): f′(1) = 2, f′(3) = 6, f′(−2) = −4, f′(0) = 0.

1. 1
   1. The increment of the function

   Write Δf = f(x₀ + Δx) − f(x₀) and simplify: substitute the whole expression x₀ + Δx for x, expand the brackets and collect like terms. After a correct simplification every term contains the factor Δx.

2. 2
   2. The ratio

   Form the ratio Δf / Δx and cancel Δx. With fractions, bring them to a common denominator first; with roots, multiply the numerator and the denominator by the conjugate expression.

3. 3
   3. The limit

   In the simplified expression let Δx → 0: now you may put 0 in place of Δx. The number you get is f′(x₀). Write x instead of x₀ and you get the function f′(x).

Differentiability and continuity

Recall: a function f is continuous at x₀ if Δf → 0 as Δx → 0, that is, a small change of the argument causes only a small change of the function. On the graph this means there is no break or jump at x₀: you can draw the graph there without lifting your pencil.

Theorem. A function that is differentiable at a point is continuous at that point. The proof takes one line: Δf = (Δf / Δx) · Δx. As Δx → 0 the first factor approaches the number f′(x₀) and the second approaches 0, so Δf → f′(x₀) · 0 = 0. The converse is false: a function can be continuous and still have no derivative. Here are two classic examples.

Finding derivatives from the definition shows you the “engine” of the derivative, but computing a limit every time is tiring. In the next lesson we study the geometric and physical meaning of the derivative and the equation of the tangent line; then come the table of derivatives and the rules for the derivative of a sum, a product and a quotient. These formulas are derived from the definition once and then used as ready-made tools.