Unit 2 · Differentiation: Definition and Fundamental Properties
● Core concept · ○ Supporting concept
2.1 Defining Average and Instantaneous Rates of Change at a Point
Secant line ○ — A line through two points on a curve; its slope is the average rate of change of the function between those points.
Tangent line ● (core concept) — The line through (c, f(c)) with slope f'(c); its slope is the instantaneous rate of change of f at c.
2.2 Defining the Derivative of a Function and Using Derivative Notation
Derivative of a function ● (core concept) — The instantaneous rate of change of f at x, defined by f'(x) = lim(h→0) [f(x+h) − f(x)] / h, provided the limit exists.
Derivative notation ● (core concept) — The derivative can be written f'(x), y', dy/dx, or d/dx[f(x)]; dy/dx is Leibniz notation and f'(x) is Lagrange (prime) notation.
2.3 Estimating Derivatives of a Function at a Point
Difference quotient ● (core concept) — The expression [f(x+h) − f(x)] / h (or [f(a+h) − f(a)] / h); its limit as h → 0 is the derivative.
2.4 Connecting Differentiability and Continuity
Differentiable ● (core concept) — A function is differentiable at x = c if f'(c) exists — equivalently, if the limit defining the derivative exists there.
Differentiability implies continuity ● (core concept) — If f is differentiable at c, then f is continuous at c. The converse is false: a continuous function can fail to be differentiable at a sharp corner, cusp, or vertical tangent.
2.5 Applying the Power Rule
Power Rule ● (core concept) — If f(x) = xⁿ, then f'(x) = n·xⁿ⁻¹.
2.6 Derivative Rules: Constant, Sum, Difference, and Constant Multiple
Constant Rule ● (core concept) — The derivative of a constant is zero: d/dx[c] = 0.
Sum Rule ● (core concept) — The derivative of a sum is the sum of the derivatives: d/dx[f + g] = f' + g'.
Difference Rule ● (core concept) — The derivative of a difference is the difference of the derivatives: d/dx[f − g] = f' − g'.
Constant Multiple Rule ● (core concept) — Constants factor out of derivatives: d/dx[c·f(x)] = c·f'(x).
2.7 Derivatives of cos x, sin x, e^x, and ln x
Derivative of sin x ● (core concept) — d/dx[sin x] = cos x.
Derivative of cos x ● (core concept) — d/dx[cos x] = −sin x.
Derivative of e^x ● (core concept) — d/dx[eˣ] = eˣ.
Derivative of ln x ● (core concept) — d/dx[ln x] = 1/x (for x > 0).
2.8 The Product Rule
Product Rule ● (core concept) — d/dx[f·g] = f'·g + f·g'.
2.9 The Quotient Rule
Quotient Rule ● (core concept) — d/dx[f/g] = (f'·g − f·g') / g².
2.10 Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions
Derivative of tan x ● (core concept) — d/dx[tan x] = sec²x.
Derivative of cot x ● (core concept) — d/dx[cot x] = −csc²x.
Derivative of sec x ● (core concept) — d/dx[sec x] = sec x·tan x.
Derivative of csc x ● (core concept) — d/dx[csc x] = −csc x·cot x.