Difficulty: Foundational | Prerequisites: Limits, continuity (Calculus I review)
The derivative is the central object in differential calculus, and everything that follows in Calculus II (integration techniques, series, parametric and polar calculus) assumes you can differentiate fluently. This set of notes covers where the derivative comes from (the limit definition), the many ways it gets written down, and the situations where it fails to exist. If you are joining the course late or feel shaky on Calc I, start here: nothing later makes sense without this.
The derivative of a function at a point is just the limit of a difference quotient. There are half a dozen notations for the same idea (Leibniz, prime, operator, dot), and you need to recognise all of them. A function fails to be differentiable wherever it is discontinuous, has a cusp, or has a vertical tangent.
Derivative
The instantaneous rate of change of a function at a point, defined as the limit of the difference quotient. In simple terms, it tells you the slope of the curve at exactly one point.
Difference quotient
The ratio [f(x + h) - f(x)] / h. Think of it as the slope of a secant line between two nearby points on the graph.
Limit definition of the derivative
f'(x) = lim(h->0) [f(x+h) - f(x)] / h. This is the formal foundation: you shrink the gap between the two secant points until they coincide.
Higher order derivative
The derivative of the derivative. The second derivative f''(x) gives the rate of change of the slope itself (concavity). Third, fourth, and nth derivatives follow the same pattern.
Differentiable (at a point)
A function is differentiable at x = a if the limit definition produces a finite value there. In simple terms, the graph is smooth enough at that point to have a single, well-defined tangent line.
Cusp
A sharp turning point where the curve abruptly changes direction. The left-hand and right-hand slopes disagree, so no single tangent line exists.
Vertical tangent
A point where the tangent line is straight up and down, meaning the slope approaches positive or negative infinity. The derivative does not exist there because it is not finite.
The derivative of f(x) at the specific point x = x₀ is:
f'(x_0) = \lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h}There is an equivalent form using a different variable:
f'(x_0) = \lim_{z \to x_0} \frac{f(z) - f(x_0)}{z - x_0}Both forms say the same thing: shrink the distance between two points on the curve until the secant line becomes the tangent line.
The key caveat: these limits must exist. If they do not, the derivative is undefined at that point.
General form (derivative as a function)
In practice, you usually compute the derivative as a function of x rather than at a single point:
f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}Once you have f'(x), you can evaluate it at any specific x = a by substitution. This saves you from reworking the limit each time.
Left-hand and right-hand derivatives work exactly like one-sided limits. When the left-hand derivative and the right-hand derivative disagree, the (two-sided) derivative does not exist.
There are many ways to write f'(x). You will see all of these used interchangeably in textbooks and on exams:
Prime notation: f'(x), y'
Leibniz notation: df/dx, dy/dx, (d/dx)[f(x)]
Operator notation: D[f(x)], Dₓ[f(x)]
Newton (dot) notation: ḟ (used mainly in physics for time derivatives)
The brackets in the operator forms can be parentheses or dropped entirely.
Higher order derivatives
Take the derivative of the derivative and you get the second derivative. Keep going for third, fourth, and nth.
Second: f''(x), y'', d²f/dx², D²f(x)
Third: f'''(x), y''', d³f/dx³, D³f(x)
Fourth and beyond: f⁽⁴⁾(x), y⁽⁴⁾, d⁴f/dx⁴ (the parentheses around the superscript number prevent confusion with exponents)
nth: f⁽ⁿ⁾(x), dⁿf/dxⁿ, Dⁿf(x)
Note: the Leibniz notation d²y/dx² is not a fraction you can cancel. The "dx²" is a single symbol meaning "with respect to x, twice."
Differentiability implies continuity. If a function is differentiable at a point, it must be continuous there. The contrapositive is the useful direction: if a function is not continuous at a point, it is automatically not differentiable there.
Beyond discontinuity, there are two other failure modes. All three are worth memorising:
Discontinuity at x = x₀. Any type of discontinuity (jump, removable, infinite) kills differentiability.
Sharp point or cusp at x = x₀. The curve changes direction abruptly. The left-hand and right-hand derivatives exist but disagree, so the two-sided limit fails. Classic example: f(x) = |x| at x = 0.
Vertical tangent at x = x₀. The slope heads to ±∞. The limit in the definition does not converge to a finite number. Example: f(x) = x^(1/3) at x = 0.
This is a real-world consideration whenever you model anything with piecewise functions, absolute values, or roots: check the join points for differentiability before assuming a derivative exists everywhere.
Students often think "continuous" and "differentiable" mean the same thing. They do not. Continuity is necessary for differentiability but not sufficient: f(x) = |x| is continuous everywhere but not differentiable at x = 0.
Students sometimes treat d²y/dx² as though you can cancel the "d"s or split the fraction. Leibniz notation looks like a fraction but does not behave like one algebraically.
Confusing the derivative at a point with the derivative as a function. f'(3) is a number (the slope at x = 3). f'(x) is a function (the slope formula for every x).
Assuming that if a function has a formula, it must be differentiable everywhere. Piecewise functions, absolute values, and nth-root functions all have formulas but can fail differentiability at specific points.
⚠️ Expect at least one question asking you to compute a derivative using the limit definition, not the shortcut rules. Show every algebraic step.
⚠️ Know all the notation forms. Exams may switch between Leibniz and prime notation within a single problem.
⚠️ "State the three ways a function can fail to be differentiable" is a standard short-answer prompt. Have the list ready: discontinuity, cusp/corner, vertical tangent.
⚠️ You may be given a graph and asked to identify where f'(x) does not exist. Look for corners, jumps, and places where the tangent goes vertical.
True or false: If f is differentiable at x = 2, then f is continuous at x = 2.
True or false: If f is continuous at x = 5, then f is differentiable at x = 5.
Fill in the blank: f'(x) = lim(h->0) ____
True or false: d²y/dx² means "the second derivative of y with respect to x."
Name one function that is continuous at x = 0 but not differentiable there.
Answers: 1. True. 2. False (e.g., |x| at 0). 3. [f(x+h) - f(x)] / h. 4. True. 5. f(x) = |x|.
Q: Use the limit definition to find f'(x) for f(x) = 3x² + 1.
A: f'(x) = lim(h->0) [(3(x+h)² + 1) - (3x² + 1)] / h = lim(h->0) [6xh + 3h²] / h = lim(h->0) (6x + 3h) = 6x.
Q: At which points is f(x) = |x - 2| not differentiable? Explain why.
A: At x = 2. The function has a sharp corner there. The left-hand derivative is -1 and the right-hand derivative is +1, so the two-sided limit does not exist.
Q: Write the third derivative of y = f(x) in Leibniz notation.
A: d³y/dx³.
Q: If g(x) = x^(2/3), is g differentiable at x = 0? Why or why not?
A: No. g'(x) = (2/3)x^(-1/3), which tends to ±∞ as x approaches 0. The graph has a vertical tangent at x = 0.
Q: True or false, and explain: "Every polynomial is differentiable on all of ℝ."
A: True. Polynomials are continuous everywhere and have no cusps or vertical tangents, so the limit definition produces a finite value at every point.
This material connects directly to the differentiation rules (product, quotient, chain) covered in the next set of notes, because those rules are all derived from the limit definition. You will also revisit the derivative when you study integration: the Fundamental Theorem of Calculus ties the derivative and the integral together as inverse operations. In Calc II, parametric differentiation and series convergence tests both rely on fluent use of derivative notation.
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