Difficulty: Introductory | Prerequisites: College algebra (functions, graphing, basic algebraic manipulation)
This block is the foundation of the entire course. Before you can do anything with derivatives or integrals, you need to understand what a limit is and what it means for a function to be continuous. Once limits are solid, the derivative follows naturally as a special kind of limit: the instantaneous rate of change. The last chunk (11.1 through 11.3) hands you the rules that let you skip the limit definition and differentiate functions quickly. If you are joining late or feel shaky, make sure your algebra (factoring, rational expressions, exponent laws) is in good shape before diving in.
Limits describe what a function approaches as the input gets close to some value. Continuity means there are no jumps or holes. The derivative measures how fast a function is changing at a single point, and a set of shortcut rules (power rule, sum rule, product rule, quotient rule) lets you compute derivatives without going back to the limit definition every time.
Limit
The value that f(x) approaches as x gets closer and closer to some number c. Written lim (x→c) f(x) = L. In simple terms, it is the answer to "where is the function heading?" even if it never arrives.
One-sided limit
The value f(x) approaches from only one direction. lim (x→c⁻) is the left-hand limit; lim (x→c⁺) is the right-hand limit. Think of it as checking the function's behaviour from only one side of the point.
Continuity
A function is continuous at x = c when three things hold: f(c) exists, lim (x→c) f(x) exists, and the limit equals f(c). In simple terms, you can draw the graph through that point without lifting your pen.
Discontinuity
A point where continuity fails. Common types: a hole (removable), a jump, or a vertical asymptote (infinite discontinuity).
Average rate of change
The slope of the secant line between two points on a graph: [f(b) - f(a)] / (b - a). Think of it as the "overall speed" between two moments, ignoring what happened in between.
Instantaneous rate of change
The slope of the tangent line at a single point. It is the limit of the average rate of change as the interval shrinks to zero. This is what the derivative gives you.
Derivative
The function f'(x) = 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 tells you how fast the output is changing right now.
Differentiable
A function is differentiable at a point if its derivative exists there. Corners, cusps, and vertical tangent lines are places where differentiability fails.
Marginal cost / marginal revenue / marginal profit
The derivative of the cost, revenue, or profit function with respect to quantity. It approximates the change in cost, revenue, or profit from producing one additional unit.
Secant line
A line passing through two points on a curve. Its slope equals the average rate of change between those points.
Tangent line
A line that touches a curve at exactly one point and has the same slope as the curve there. Its slope equals the derivative at that point.
To estimate a limit numerically, build a table of x-values approaching c from both sides and watch what f(x) approaches.
On a graph, trace the curve toward x = c from the left and from the right. If both sides head to the same y-value, that is the limit.
The limit can exist even if f(c) itself is undefined (a hole in the graph).
If the left-hand and right-hand limits disagree, the two-sided limit does not exist.
A function is continuous at x = c when: f(c) is defined, lim (x→c) f(x) exists, and lim (x→c) f(x) = f(c).
Polynomials are continuous everywhere. Rational functions are continuous everywhere except where the denominator is zero.
Removable discontinuity (hole): the limit exists but f(c) is either undefined or differs from the limit. You can "fix" it by redefining f(c).
Jump discontinuity: left-hand and right-hand limits both exist but are not equal.
Infinite discontinuity: the function blows up to ±∞ near c (vertical asymptote).
Direct substitution: plug c into f(x). If you get a finite number, that is the limit.
If substitution gives 0/0 (indeterminate form), simplify first. Typical moves:
Factor and cancel common terms.
Multiply by a conjugate (useful when square roots appear).
Simplify a complex fraction.
After cancelling, substitute again to get the limit.
Limits at infinity: divide numerator and denominator by the highest power of x in the denominator. Terms with x in the denominator vanish, leaving the ratio of leading coefficients (or 0 or ±∞ depending on degrees).
Average rate of change of f from x = a to x = b: [f(b) - f(a)] / (b - a).
Geometrically, this is the slope of the secant line through (a, f(a)) and (b, f(b)).
Units: (units of output) per (units of input). If f measures dollars and x measures items, the average rate of change is dollars per item.
The derivative at x = a is the limit of the average rate of change as the interval shrinks: f'(a) = lim (h→0) [f(a+h) - f(a)] / h.
Graphically, the derivative is the slope of the tangent line at x = a.
You can estimate f'(a) numerically using a small h (balanced difference quotient: [f(a+h) - f(a-h)] / 2h is often more accurate).
Where the tangent is flat (horizontal), f'(a) = 0.
Where the graph is rising, f'(a) > 0. Where it is falling, f'(a) < 0.
Use the limit definition to compute f'(x) as a formula:
Write the difference quotient [f(x+h) - f(x)] / h.
Expand and simplify.
Cancel h from the numerator and denominator.
Take the limit as h→0.
The result is a new function, f'(x), that gives the slope of the tangent line at any x in its domain.
The equation of the tangent line at x = a is: y - f(a) = f'(a)(x - a).
Constant rule: If f(x) = c (a constant), then f'(x) = 0.
Power rule: If f(x) = xⁿ, then f'(x) = nxⁿ⁻¹. Works for any real exponent, including negatives and fractions.
Constant multiple rule: d/dx [c · f(x)] = c · f'(x). Pull constants out front, then differentiate.
Sum/difference rule: d/dx [f(x) ± g(x)] = f'(x) ± g'(x). Differentiate term by term.
Rewrite before differentiating: convert roots to fractional exponents (√x = x^(1/2)) and move terms out of denominators (1/x² = x⁻²).
In business calculus, the cost function C(x), revenue function R(x), and profit function P(x) = R(x) - C(x) all depend on quantity x.
Marginal cost = C'(x), the approximate cost of the next unit. Marginal revenue = R'(x). Marginal profit = P'(x).
At the profit-maximising quantity, P'(x) = 0, which means R'(x) = C'(x): marginal revenue equals marginal cost.
"Marginal" does not mean "small." It means "derivative of." The marginal cost at x = 100 approximates C(101) - C(100).
Product rule: d/dx [f(x) · g(x)] = f'(x) · g(x) + f(x) · g'(x). Remember: "derivative of the first times the second, plus the first times the derivative of the second."
Quotient rule: d/dx [f(x) / g(x)] = [f'(x) · g(x) - f(x) · g'(x)] / [g(x)]². Remember: "lo d-hi minus hi d-lo, over lo-lo" (lo = denominator, hi = numerator).
You can often avoid the quotient rule by rewriting f/g as f · g⁻¹ and using the product rule with the power rule on g⁻¹. Both approaches give the same result.
Apply these rules in combination when a function has more than two factors or nested quotients.
Limit definition of the derivative: f'(x) = lim (h→0) [f(x+h) - f(x)] / h
Power rule: d/dx [xⁿ] = nxⁿ⁻¹
Constant rule: d/dx [c] = 0
Constant multiple rule: d/dx [c · f(x)] = c · f'(x)
Sum/difference rule: d/dx [f ± g] = f' ± g'
Product rule: d/dx [f · g] = f' · g + f · g'
Quotient rule: d/dx [f / g] = [f' · g - f · g'] / g²
Average rate of change: [f(b) - f(a)] / (b - a)
Tangent line at x = a: y = f(a) + f'(a)(x - a)
The derivative is the mathematical tool behind any question of the form "how fast is something changing?" Economists use marginal cost and marginal revenue (derivatives of cost and revenue functions) to decide optimal production levels. In physics, velocity is the derivative of position with respect to time, and acceleration is the derivative of velocity. Even something as routine as a speedometer reading is an instantaneous rate of change.
Students often think a limit equals the function value at that point. It does not. The limit is about what f(x) approaches, not what f(c) equals. A function can have a limit of 5 at x = 2 even if f(2) is undefined.
Students confuse "the limit does not exist" with "the function is undefined." These are separate ideas. A function can be undefined at a point yet still have a limit there (a hole). Conversely, f(c) can be defined while the limit fails to exist (a jump).
When the difference quotient gives 0/0, students sometimes conclude the limit is 0 or undefined. The 0/0 form is indeterminate, meaning you need to simplify further before evaluating.
Students mix up the product rule and the "multiply the derivatives" shortcut (which does not exist). d/dx [f · g] is not f' · g'. You must use f' · g + f · g'.
⚠️ Expect at least one problem requiring the limit definition of the derivative (the long way, not the shortcut rules). Know every step.
⚠️ Be able to determine continuity at a point by checking all three conditions, not just one.
⚠️ Algebraic limit problems (factor and cancel, conjugate multiplication) appear on virtually every exam.
⚠️ Marginal analysis word problems are a staple: given a cost or revenue function, find and interpret the derivative at a specific quantity.
⚠️ Know when and how to apply each differentiation rule. The product and quotient rules are tested in combination, not in isolation.
True or false: If f(3) = 7, then lim (x→3) f(x) must also equal 7.
Fill in the blank: The derivative of x⁴ is ____.
True or false: The quotient rule states d/dx [f/g] = f'/g'.
Fill in the blank: If C(x) is a cost function, then C'(x) is called the ____ cost.
True or false: A function can be continuous at a point but not differentiable there.
Answers: 1. False (the limit depends on behaviour near 3, not at 3). 2. 4x³. 3. False (it is [f'g - fg'] / g²). 4. Marginal. 5. True (example: f(x) = |x| at x = 0).
Q: Find lim (x→2) (x² - 4) / (x - 2).
A: Factor the numerator: (x² - 4) = (x - 2)(x + 2). Cancel (x - 2). The limit becomes lim (x→2) (x + 2) = 4.
Q: Is f(x) = (x² - 9) / (x - 3) continuous at x = 3? Why or why not?
A: No. f(3) is undefined because the denominator is zero. The limit exists (it equals 6, after cancelling), but continuity requires f(c) to be defined.
Q: Use the limit definition to find the derivative of f(x) = 3x².
A: [f(x+h) - f(x)] / h = [3(x+h)² - 3x²] / h = [3x² + 6xh + 3h² - 3x²] / h = 6x + 3h. As h→0, f'(x) = 6x.
Q: A company's cost function is C(x) = 0.5x² + 10x + 200. Find the marginal cost at x = 50 and interpret it.
A: C'(x) = x + 10, so C'(50) = 60. This means the approximate cost of producing the 51st unit is $60.
Q: Differentiate f(x) = (2x + 1)(x³ - x) using the product rule.
A: f'(x) = 2(x³ - x) + (2x + 1)(3x² - 1) = 2x³ - 2x + 6x³ - 2x + 3x² - 1 = 8x³ + 3x² - 4x - 1.
Limits are the language the rest of the course speaks. Every derivative is defined as a limit, and later, every definite integral will be defined as a limit of sums (Riemann sums). Continuity matters again when you reach the Fundamental Theorem of Calculus, which requires the integrand to be continuous on the interval. The differentiation rules from 11.1 through 11.3 are extended by the chain rule (11.4), which handles compositions of functions, and by the rules for logarithmic and exponential functions (11.5).
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