Source: Calculus for Business, Chapter 1 (FIU)
Tags: derivative, difference quotient, secant line, tangent line, slope, rate of change, limit definition of derivative
Difficulty: Introductory | Prerequisites: Limits and Continuity study notes (the first document in this set). You need to be comfortable evaluating limits before tackling derivative definitions.
The derivative is the central tool of calculus. It tells you the instantaneous rate of change of a function at any point, which in a business context translates to marginal cost, marginal revenue, or the rate at which profit is changing. This section of Chapter 1 bridges the limit material you have just covered with the formal definition of the derivative. Everything here is done "the long way" using limits; the shortcut rules (power rule, product rule, etc.) come later. Understand this definition first, because exam questions will ask you to use it directly.
The derivative of f(x) is the limit of the difference quotient [f(x+h) − f(x)] / h as h → 0. Geometrically, it is the slope of the tangent line at a point. This chapter derives it from scratch for simple functions like f(x) = x and f(x) = x², using the same limit techniques from the first half of the chapter.
Secant line
A straight line that passes through two points on the graph of a function. Its slope gives the average rate of change between those two points.
Think of it as: the line connecting two dots on a curve.
Tangent line
A straight line that touches the graph of a function at exactly one point (locally) and has the same slope as the curve at that point.
Think of it as: the line that "just grazes" the curve. It is what the secant line becomes when the two points merge into one.
Difference quotient
The expression [f(x+h) − f(x)] / h. It measures the average rate of change of f over the interval from x to x + h.
In simple terms, this is "rise over run" between two nearby points on the curve.
Derivative
The function f'(x) defined as lim (h→0) [f(x+h) − f(x)] / h. It gives the instantaneous rate of change of f at any point x.
In simple terms, the derivative is what the average rate of change becomes when the gap between the two points shrinks to zero.
f'(x) (f prime of x)
Notation for the derivative of f with respect to x. Also written as dy/dx when y = f(x).
Think of it as: the slope of the tangent line to f at the point x.
From two points to a slope
Pick two points on the graph of f: (x, f(x)) and (x₀, f(x₀)).
The slope of the secant line through them is Δy / Δx = [f(x₀) − f(x)] / (x₀ − x).
This is the average rate of change of f between x and x₀.
Moving the second point closer
Now let the second point slide towards the first. Replace x₀ with x + h, so the slope becomes [f(x+h) − f(x)] / h.
As h gets smaller, the secant line rotates towards the tangent line.
In the limit (h → 0), the secant becomes the tangent, and its slope becomes the derivative.
The derivative of f(x) with respect to x is the function f'(x) given by:
f'(x) = lim (h→0) [f(x+h) − f(x)] / h
provided this limit exists.
This is sometimes called the "limit definition of the derivative" or the "definition from first principles." On exams, when a question says "use the definition" or "use the limit process," this is what they mean. You are not allowed to use shortcut rules (power rule, etc.) for these questions.
Example 1: f(x) = x
f'(x) = lim (h→0) [f(x+h) − f(x)] / h
= lim (h→0) [(x+h) − x] / h
= lim (h→0) h / h
= lim (h→0) 1
= 1
The derivative of f(x) = x is 1. The graph of y = x is a straight line with slope 1, which makes sense: the tangent line to a straight line is the line itself.
Example 2: f(x) = x²
f'(x) = lim (h→0) [f(x+h) − f(x)] / h
= lim (h→0) [(x+h)² − x²] / h
= lim (h→0) [x² + 2xh + h² − x²] / h
= lim (h→0) [2xh + h²] / h
= lim (h→0) h(2x + h) / h
= lim (h→0) (2x + h)
= 2x
The derivative of f(x) = x² is f'(x) = 2x. At x = 3, the slope of the tangent is 6. At x = −1, the slope is −2.
The general pattern for these problems
Write out the definition: lim (h→0) [f(x+h) − f(x)] / h.
Substitute f(x+h) by replacing every x in the formula with (x+h).
Expand and simplify the numerator.
Factor out h from the numerator so it cancels with the h in the denominator.
Take the limit as h → 0 by direct substitution.
Difference quotient (average rate of change)
\frac{\Delta y}{\Delta x} = \frac{f(x_0) - f(x)}{x_0 - x}Limit definition of the derivative
f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}Results derived in this chapter
f(x) = x \implies f'(x) = 1f(x) = x^2 \implies f'(x) = 2xIn business, the derivative of a cost function C(x) gives the marginal cost: the approximate cost of producing one additional unit. The derivative of a revenue function R(x) gives marginal revenue. Profit is maximised where marginal revenue equals marginal cost, which is a derivative equation. Every optimisation problem in this course rests on the definition you are learning here.
Students often try to plug h = 0 into the difference quotient immediately and get 0/0. You must simplify algebraically first so the h in the denominator cancels, then take the limit.
Students confuse the secant line with the tangent line. The secant passes through two points; the tangent is the limiting position as those points merge.
Students sometimes think f'(x) is a single number. It is a function of x. It gives a different slope at each point along the curve.
Students drop the "lim" notation partway through the calculation. Keep writing lim (h→0) on every line until you actually set h = 0 in the final step. Dropping it early is a notation error that loses marks.
⚠️ "Find f'(x) using the limit definition" is a guaranteed exam question in this course. Practise the algebra until it is automatic.
⚠️ You will need to expand (x+h)² and sometimes (x+h)³ cleanly. Errors in expansion are the most common source of wrong answers.
⚠️ Know the geometric interpretation: the derivative is the slope of the tangent line. Questions may ask you to find the equation of the tangent line at a given point.
⚠️ The step where you factor h out of the numerator and cancel is where most students get stuck. If you cannot cancel h, something went wrong in the expansion.
True or False: The difference quotient gives the instantaneous rate of change. (False, it gives the average rate of change. The derivative, which is the limit of the difference quotient, gives the instantaneous rate.)
Fill in the blank: f'(x) = lim (h→0) ___. ([f(x+h) − f(x)] / h)
True or False: The derivative of f(x) = x² is 2x. (True)
Fill in the blank: The slope of the secant line through (x, f(x)) and (x+h, f(x+h)) is ___. ([f(x+h) − f(x)] / h)
True or False: If f'(3) = 6, the tangent line to f at x = 3 has slope 6. (True)
Q: Use the limit definition to find f'(x) for f(x) = 3x.
A: f'(x) = lim (h→0) [3(x+h) − 3x] / h = lim (h→0) 3h / h = lim (h→0) 3 = 3.
Q: Use the limit definition to find f'(x) for f(x) = x² + 1.
A: f'(x) = lim (h→0) [(x+h)² + 1 − (x² + 1)] / h = lim (h→0) [x² + 2xh + h² + 1 − x² − 1] / h = lim (h→0) [2xh + h²] / h = lim (h→0) (2x + h) = 2x.
Q: What is the slope of the tangent line to f(x) = x² at x = 4?
A: f'(x) = 2x, so f'(4) = 8. The tangent line at x = 4 has slope 8.
Q: Explain, in one sentence, the geometric relationship between a secant line and a tangent line.
A: The tangent line is the limiting position of the secant line as the two points on the curve are brought together.
Q: A student writes f'(x) = [f(x+h) − f(x)] / h and then immediately substitutes h = 0 to get 0/0. What should they do instead?
A: They should first expand and simplify the numerator, factor out h, cancel with the denominator, and only then take the limit as h → 0.
The limit definition of the derivative is the foundation for every differentiation rule that follows: the power rule, product rule, quotient rule, and chain rule are all proved by starting here and finding patterns. In business calculus, derivatives are the tool for all optimisation problems (maximising profit, minimising cost) and for marginal analysis. This connects directly to Chapter 2's differentiation rules and Chapter 3's applications of the derivative.
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