Difficulty: Intermediate | Prerequisites: Limits (Ch. 1 Sec. 4–6), continuity basics.
Chapter 2 turns the limit machinery of Chapter 1 into the central tool of calculus: the derivative. Sections 1 and 2 define the derivative (first at a single point, then as a full function) using the limit of the difference quotient, and catalogue the ways a function can fail to be differentiable. Section 3 replaces the limit definition with shortcut rules (power, sum/difference, product, quotient) that make differentiation fast.
The derivative of a function at a point is the slope of the tangent line there, defined as the limit of the difference quotient. Once you understand the definition, shortcut rules (power, product, quotient) let you differentiate almost any expression without going back to limits every time.
Derivative of f at c, f'(c)
The number f'(c) = lim(h→0) [f(c + h) - f(c)] / h, provided the limit exists. It gives the instantaneous rate of change of f at x = c and the slope of the tangent line to y = f(x) at (c, f(c)).
Derivative as a function, f'(x)
The function f'(x) = lim(h→0) [f(x + h) - f(x)] / h, defined at every x where this limit exists. It is the "slope function" that tells you the slope of the tangent line at each point.
Leibniz notation
Writing df/dx or (d/dx) f(x) instead of f'(x). The notation df/dx|_{x=c} means the derivative evaluated at x = c. It looks like a fraction but is really a single symbol for the derivative.
Differentiable
A function is differentiable at c if f'(c) exists. Differentiable on an interval means differentiable at every point in that interval. If a function is differentiable at c, it must be continuous there (but continuity does not guarantee differentiability).
Power rule
If f(x) = xⁿ for any real number n, then f'(x) = n xⁿ⁻¹. This single rule handles polynomials, roots (write √x as x^(1/2)), and reciprocals (write 1/x³ as x⁻³).
Constant rule
The derivative of a constant is 0. The derivative of c · f(x) is c · f'(x).
Sum and difference rules
(f + g)' = f' + g' and (f - g)' = f' - g'. Derivatives distribute over addition and subtraction.
Product rule
(f · g)' = f' · g + f · g'. In words: derivative of the first times the second, plus the first times the derivative of the second.
Quotient rule
(f/g)' = [f' · g - f · g'] / g². A common mnemonic: "low d-high minus high d-low, all over low squared."
At a specific point c: f'(c) = lim(h→0) [f(c + h) - f(c)] / h. Equivalent form: f'(c) = lim(x→c) [f(x) - f(c)] / (x - c).
As a function: f'(x) = lim(h→0) [f(x + h) - f(x)] / h.
To use the definition: write out f(x + h), expand it, subtract f(x), divide by h, simplify, then take the limit as h → 0.
Example: f(x) = x² + x. Then f(x + h) = (x + h)² + (x + h) = x² + 2xh + h² + x + h. So [f(x + h) - f(x)] / h = (2xh + h² + h) / h = 2x + h + 1. As h → 0, f'(x) = 2x + 1.
Once you know f'(c), the tangent line at (c, f(c)) is: y - f(c) = f'(c)(x - c).
This is point-slope form with slope m = f'(c) and point (c, f(c)).
Three main cases where f'(c) fails to exist:
Cusp or corner (jagged edge): The left-hand and right-hand derivatives (one-sided limits of the difference quotient) disagree. Example: f(x) = |x - 1| at x = 1. The slope from the left is -1, from the right is +1.
Discontinuity: If f is not continuous at c, it cannot be differentiable there. Differentiability implies continuity (Theorem 2.6), but not the other way round.
Vertical tangent: The tangent line is vertical, so the slope is undefined (the limit of the difference quotient is ±∞). Example: f(x) = ∛x at x = 0.
Where f has a horizontal tangent (local max or min), f' = 0.
Where f is increasing, f' > 0. Where f is decreasing, f' < 0.
Estimate slopes at several points and plot them to sketch f'(x).
Constants and lines:
d/dx [c] = 0. A horizontal line has zero slope.
d/dx [mx + b] = m. The slope of a line is constant.
Power rule: d/dx [xⁿ] = n xⁿ⁻¹ for any real n.
Rewrite roots and reciprocals as powers first: √x = x^(1/2), 1/x³ = x⁻³, ∛x = x^(1/3).
Example: d/dx [x^(5/2)] = (5/2) x^(3/2).
Example: d/dx [1/x] = d/dx [x⁻¹] = -x⁻² = -1/x².
Sum and difference: d/dx [f ± g] = f' ± g'. Derivatives pass through addition and subtraction.
Constant multiple: d/dx [c · f] = c · f'.
Product rule: d/dx [f · g] = f' · g + f · g'.
Example: differentiate (x² - 3x)(√x + x + 3). Apply the product rule: (2x - 3)(√x + x + 3) + (x² - 3x)(1/(2√x) + 1).
Quotient rule: d/dx [f/g] = [f' · g - f · g'] / g².
Example: find the tangent line to y = 2x/(x + 1) at (1, 1). y' = [(2)(x + 1) - (2x)(1)] / (x + 1)² = 2/(x + 1)². At x = 1: y' = 2/4 = 1/2. Tangent line: y - 1 = (1/2)(x - 1).
The second derivative f''(x) is the derivative of f'(x). In Leibniz notation: d²f/dx².
The third derivative is f'''(x), and so on.
Example: g(x) = -3x⁵ + 12x² + 10. g'(x) = -15x⁴ + 24x. g''(x) = -60x³ + 24. g'''(x) = -180x².
f'(c) = \lim_{h \to 0} \frac{f(c+h) - f(c)}{h}\frac{d}{dx}[x^n] = n\,x^{n-1} \qquad \frac{d}{dx}[c] = 0 \qquad \frac{d}{dx}[cf] = c\,f'(f \cdot g)' = f'g + fg' \qquad \left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}Tangent line at (c, f(c)): y - f(c) = f'(c)(x - c).
Students assume (f · g)' = f' · g'. This is wrong. The product rule has two terms: f'g + fg'. There is no shortcut that lets you just differentiate each factor separately.
Similarly, (f/g)' ≠ f'/g'. The quotient rule is more involved and includes a minus sign and a denominator squared.
Forgetting to rewrite roots and fractions as powers before applying the power rule. If you leave √x as √x, the power rule is awkward to apply. Write it as x^(1/2) first.
Confusing differentiability with continuity. Every differentiable function is continuous, but |x| is continuous at 0 without being differentiable there.
⚠️ The limit definition of the derivative is a major learning objective. Expect at least one exam question requiring you to find a derivative from scratch using lim(h→0) [f(x+h) - f(x)] / h, not the shortcut rules.
⚠️ The product and quotient rules are tested constantly from this point onward. Memorise them cold and practise until applying them is automatic.
⚠️ Tangent line problems: compute f'(c) for the slope, evaluate f(c) for the point, write point-slope form. This appears on virtually every exam.
⚠️ Know the three cases where a derivative fails to exist (cusp, discontinuity, vertical tangent) and be able to identify them from a graph.
True or False: If f is continuous at x = 3, then f is differentiable at x = 3.
Fill in the blank: d/dx [x⁷] = ___.
True or False: (f · g)' = f' · g + f · g'.
Fill in the blank: d/dx [5/x²] = d/dx [5x⁻²] = ___.
True or False: The derivative of a constant is 1.
Answers: 1. False (continuity does not guarantee differentiability). 2. 7x⁶. 3. True (product rule). 4. -10x⁻³ = -10/x³. 5. False (it is 0).
Q: Use the limit definition to find f'(3) for f(x) = 2x + 5.
A: [f(3 + h) - f(3)] / h = [2(3 + h) + 5 - 11] / h = 2h/h = 2. So f'(3) = 2.
Q: Compute f'(x) using the limit definition for f(x) = x² + x - 5.
A: f(x + h) = (x + h)² + (x + h) - 5 = x² + 2xh + h² + x + h - 5. [f(x + h) - f(x)] / h = (2xh + h² + h)/h = 2x + h + 1. As h → 0, f'(x) = 2x + 1.
Q: Find the equation of the tangent line to g(x) = √x at x = 1.
A: g(1) = 1, g'(x) = (1/2)x^(-1/2), g'(1) = 1/2. Tangent line: y - 1 = (1/2)(x - 1), or y = (1/2)x + 1/2.
Q: Differentiate h(x) = (x² + 4x - 32)(x⁴ + 7).
A: Product rule: h'(x) = (2x + 4)(x⁴ + 7) + (x² + 4x - 32)(4x³).
Q: Find the tangent line to y = 2x/(x + 1) at (1, 1).
A: y' = [2(x + 1) - 2x(1)]/(x + 1)² = 2/(x + 1)². At x = 1: y' = 1/2. Tangent: y - 1 = (1/2)(x - 1).
The limit definition here is the formal version of the tangent-line idea from Chapter 1, Section 4. The differentiation rules in Section 3 will be extended by the chain rule (Section 2.5), trig derivatives (Section 2.4), and implicit differentiation (Section 2.6). The product and quotient rules reappear in every application problem for the rest of the course.
derivative, definition of derivative, difference quotient, limit definition, f prime, Leibniz notation, dy/dx, tangent line equation, point-slope form, differentiable, non-differentiable, cusp, corner, vertical tangent, power rule, constant rule, sum rule, difference rule, product rule, quotient rule, higher-order derivative, second derivative, MTH 132, Calculus I, Chapter 2, Sections 2.1, 2.2, 2.3, Stewart Calculus