Differentiation: Definition and Fundamental Properties

Ten topics that turn the limit into a machine. Average rate on an interval against the instantaneous rate at a point, the difference quotient and the two limit definitions it produces, notation that moves between a value and a function, estimating a derivative from data, the four ways a derivative fails to exist, and the rules: power, constant multiple, sum and difference, then product and quotient, plus the derivatives of sine, cosine, e^x and ln x, plus the four remaining trigonometric functions.

AB exam 10-15%BC exam 5-10%10 topics

Topics

2.1

Defining Average and Instantaneous Rates of Change at a Point

Defining the Derivative of a Function and Using Derivative Notation

Estimating Derivatives of a Function at a Point

Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist

Applying the Power Rule

Derivative Rules: Constant, Sum, Difference, and Constant Multiple

Derivatives of cos x, sin x, e^x, and ln x

The Product Rule

The Quotient Rule

Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions

Lesson

Diagnostic

after diagnostic

Key forms For every problem in this unit

Average rate on [a, b]

(f(b) − f(a)) / (b − a), the secant slope

Derivative at a (h form)

limit of (f(a + h) − f(a)) / h as h → 0

Derivative at a (x form)

limit of (f(x) − f(a)) / (x − a) as x → a

Derivative function

f′(x): limit of (f(x + h) − f(x)) / h as h → 0

Notation, same object

f′(a) = dy/dx at x = a = d/dx[f(x)] at x = a

Symmetric estimate from data

f′(a) ≈ (f(a + h) − f(a − h)) / (2h)

Tangent line at a

y = f(a) + f′(a)(x − a)

Differentiable ⇒ continuous

the converse is FALSE: |x| at 0

No derivative at

corner, cusp, vertical tangent, discontinuity

Power

d/dx[xⁿ] = n·xⁿ⁻¹, for real n, WHEREVER xⁿ is real-valued and differentiable

Constant

d/dx[c] = 0, not c·x⁰

Constant multiple & sum

(cf)′ = c·f′; (f ± g)′ = f′ ± g′

Product

(uv)′ = u′v + uv′, NOT u′v′

Quotient

(u/v)′ = (u′v − uv′) / v², order matters

Sine & cosine

(sin x)′ = cos x; (cos x)′ = −sin x

Exponential

(e^x)′ = e^x; (a^x)′ = a^x·ln a

Logarithm

(ln x)′ = 1/x; (log_bx)′ = 1 / (x·ln b)

Tangent & cotangent

(tan x)′ = sec²x; (cot x)′ = −csc²x

Secant & cosecant

(sec x)′ = sec x tan x; (csc x)′ = −csc x cot x

Co-function pattern

every co- function carries the minus sign

Unit 2 tools

Challenge bank

1 / 60

60 open-ended problems.

Read the question, work it out, then flip the card to compare your reasoning to the worked solution. Mark each card so you can return to the ones that still bite.

Question

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Worked solution

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Cumulative assessment

Test the unit.

Twenty mixed items drawn from across all 10 topics, with guaranteed misconception-code coverage. Identifies which misconceptions still bite when you cannot see which topic the question came from.

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Check what stuck.

Units 1 through 2, drawn evenly so earlier units get the same share as this one. Twenty questions or a full 45-question section, your choice. Even coverage means this is a retention check rather than a score estimate. Follows the track you chose on the course page.

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26topics

30codes covered

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