Differentiation: Definition and Fundamental Properties

Unit Overview

Unit 1 ended with the idea that the instantaneous rate of change is a limit of secant slopes. This unit turns that idea into a precise, computable object: the derivative. You will define the derivative as a limit, then build a toolkit of rules — power, product, quotient, and the derivatives of the basic trig functions — that lets you differentiate quickly without returning to the limit each time.

Differentiation is the single most-used skill in AP Calculus. Almost every later unit assumes you can differentiate fluently and correctly. This unit is about 10–12% of the exam on its own, but its reach is total.

Big picture: The derivative f'(x) measures how fast f changes, and geometrically it is the slope of the tangent line.

Learning Objectives

You will be able to:

• Define f'(x) using the limit of a difference quotient (both forms).
• Interpret the derivative as a slope of a tangent line and an instantaneous rate of change.
• Estimate derivatives from graphs and tables.
• Explain the relationship between differentiability and continuity (and where differentiability fails).
• Apply the Power Rule, constant multiple, and sum/difference rules.
• Differentiate sin, cos, tan, cot, sec, csc.
• Apply the Product Rule and Quotient Rule.
• Compute higher-order derivatives.
• Perform implicit differentiation and differentiate inverse functions.

Prerequisite Concepts

• Limits and the difference quotient (Unit 1).
• Exponent rules, including negative and fractional exponents.
• Trig values and identities.
• Function composition and the meaning of an inverse function.

Key Definitions

• Difference quotient: (f(x + h) − f(x)) / h — the average rate of change over a step of size h.
• Derivative (limit definition): f'(x) = lim_{h→0} (f(x + h) − f(x)) / h
• Alternate (point) form: f'(a) = lim_{x→a} (f(x) − f(a)) / (x − a)
• Notations: f'(x), dy/dx, y', d/dx[f(x)], and for higher orders f''(x), d²y/dx².
• Differentiable at a: the limit defining f'(a) exists (the graph has a single, non-vertical tangent line there).
• Higher-order derivative: the derivative of a derivative; f'' is the second derivative, f''' the third, etc.

Important Theorems

Differentiability implies continuity. If f is differentiable at a, then f is continuous at a. The converse is false: a function can be continuous but not differentiable (e.g., |x| at x = 0).

Where differentiability fails: at corners (like |x|), cusps (like x^(2/3) at 0), vertical tangents, and any discontinuity.

Inverse Function Derivative Theorem. If f is differentiable and one-to-one with inverse g, then g'(a) = 1 / f'(g(a)), provided f'(g(a)) ≠ 0.

Important Formulas

Basic rules (c constant, n any real number):

• Constant: d/dx[c] = 0
• Power Rule: d/dx[x^n] = n·x^(n−1)
• Constant multiple: d/dx[c·f] = c·f'
• Sum/Difference: 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 (remember: "low d-high minus high d-low, over low squared")

Trig derivatives:

• d/dx[sin x] = cos x
• d/dx[cos x] = −sin x
• d/dx[tan x] = sec² x
• d/dx[cot x] = −csc² x
• d/dx[sec x] = sec x · tan x
• d/dx[csc x] = −csc x · cot x

Concept Explanations

Two forms of the derivative (2.2). The h → 0 form gives the derivative function f'(x); the x → a form gives the derivative value f'(a) directly. Recognizing a limit as one of these forms is a common AP multiple-choice task: e.g., lim_{h→0} ((2+h)⁴ − 16)/h is exactly f'(2) for f(x) = x⁴, which is 32.

Differentiability vs. continuity (2.4). Differentiability is stronger than continuity. To be differentiable, a function must be continuous and smooth (no corner, cusp, or vertical tangent). For a piecewise function, check both that the pieces meet (continuity) and that their one-sided derivatives match (smoothness).

Choosing a rule (2.11). Read the structure of the expression:

• A single power → Power Rule.
• A product of two functions of x → Product Rule.
• A quotient → Quotient Rule.
• (A function inside another → Chain Rule, formally Unit 3, but often needed alongside these.)

Implicit differentiation (2.13). When y is defined implicitly (e.g., x² + y² = 25), differentiate both sides with respect to x, treating y as a function of x (so d/dx[y²] = 2y·(dy/dx)), then solve for dy/dx.

Inverse-function derivatives (2.14). You do not need a formula for the inverse itself. Use g'(a) = 1 / f'(g(a)): find the input where f outputs a, then reciprocate the derivative of f there.

Worked Examples

Example 1 (Limit definition). Find f'(x) for f(x) = x² from the definition. lim_{h→0} ((x+h)² − x²)/h = lim_{h→0} (2xh + h²)/h = lim_{h→0} (2x + h) = 2x.

Example 2 (Power Rule with fractional/negative exponents).

• d/dx[x^(3/2)] = (3/2)x^(1/2).
• d/dx[4/x²] = d/dx[4x^(−2)] = −8x^(−3) = −8/x³.

Example 3 (Product Rule). d/dx[x² sin x] = 2x·sin x + x²·cos x.

Example 4 (Quotient Rule). d/dx[(2x + 1)/(x − 1)] = (2(x−1) − (2x+1)(1))/(x−1)² = −3/(x−1)².

Example 5 (Higher-order). For f(x) = x⁵: f'(x) = 5x⁴, f''(x) = 20x³, f'''(x) = 60x².

Example 6 (Implicit). For x² + y² = 25: differentiate to get 2x + 2y·(dy/dx) = 0, so dy/dx = −x/y. At (3, 4), the tangent slope is −3/4.

Example 7 (Inverse). For f(x) = x³ with inverse g, find g'(8). Since 2³ = 8, g(8) = 2. f'(x) = 3x², so f'(2) = 12. Thus g'(8) = 1/12.

Graphical Interpretation

• f'(x) is the slope of the tangent to y = f(x) at x.
• Where f is increasing, f' > 0; where decreasing, f' < 0; at a smooth peak/valley, f' = 0.
• A corner ⇒ left and right tangent slopes differ ⇒ not differentiable.
• A vertical tangent ⇒ f'(x) → ±∞ ⇒ not differentiable, even though the function may be continuous.
• The graph of f' records the slopes of f; reading one from the other is a frequent exam skill (developed further in Unit 5).

Common Mistakes

• Quotient Rule sign/order errors. It is (low·d-high − high·d-low)/low², not the reverse — the subtraction order matters.
• Forgetting the Product Rule and writing d/dx[f·g] = f'·g' (wrong).
• Dropping the dy/dx factor in implicit differentiation when differentiating a y-term.
• Assuming continuity ⇒ differentiability. |x| is the classic counterexample at 0.
• Power Rule on the wrong form. Rewrite roots and reciprocals as exponents first (√x = x^(1/2), 1/x² = x^(−2)).
• Inverse-derivative confusion. g'(a) = 1/f'(g(a)), not 1/f'(a).

AP Exam Tips

• Recognize limit definitions of the derivative in disguise — they are common multiple-choice items.
• Know the six trig derivatives cold; the "co-" functions (cos, cot, csc) all carry a negative sign.
• On implicit differentiation free-response, show the dy/dx factor explicitly — it earns method points.
• For a tangent-line question: find the point and the slope f'(a), then use point-slope form.
• Simplify only as needed; an unsimplified but correct derivative still earns full credit on free-response.

Unit Summary

• The derivative is f'(x) = lim_{h→0} (f(x+h) − f(x))/h — a slope of a tangent and an instantaneous rate.
• Differentiability ⇒ continuity, but not conversely (corners, cusps, vertical tangents break differentiability).
• Master the Power, Product, Quotient rules and the six trig derivatives.
• Implicit differentiation differentiates both sides in x (with dy/dx on y-terms); inverse derivatives use g'(a) = 1/f'(g(a)).
• Higher-order derivatives are derivatives of derivatives.

Next unit: the Chain Rule and its combinations (composite, implicit, and inverse functions).