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Unit 2: Differentiation: Definition and Fundamental Properties

10 dot points across 10 inquiry questions. Click any dot point for a focused answer with worked past exam questions where available.

How does the average rate of change over an interval become the instantaneous rate of change at a point?

Topic 2.1 Defining Average and Instantaneous Rates of Change at a Point: compute the average rate of change over an interval and define the instantaneous rate of change as the limit of average rates.
A focused answer to AP Calculus AB Topic 2.1, defining average rate of change as a secant slope and the instantaneous rate as its limit (the derivative), with worked secant-to-tangent examples.
8 min answer →

How do the constant, sum, difference, and constant-multiple rules let you differentiate any polynomial term by term?

Topic 2.6 Derivative Rules: Constant, Sum, Difference, and Constant Multiple: apply the basic linearity rules of differentiation to combine derivatives of individual terms.
A focused answer to AP Calculus AB Topic 2.6, covering the constant rule, constant-multiple rule, and sum and difference rules that let you differentiate polynomials term by term, with worked examples.
8 min answer →

What is the formal definition of the derivative, and how do you write it in the different standard notations?

Topic 2.2 Defining the Derivative of a Function and Using Derivative Notation: state the limit definition of the derivative, compute derivatives from the definition, and use standard derivative notation.
A focused answer to AP Calculus AB Topic 2.2, giving the two limit definitions of the derivative, the standard notations, and how to differentiate from first principles, with full worked examples.
9 min answer →

How do the derivatives of tangent, cotangent, secant, and cosecant follow from sine and cosine and the quotient rule?

Topic 2.10 Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions: derive and apply the derivatives of the remaining trigonometric functions.
A focused answer to AP Calculus AB Topic 2.10, deriving the derivatives of tangent, cotangent, secant and cosecant from sine and cosine via the quotient rule, with the full table and worked examples.
8 min answer →

What are the derivatives of the core transcendental functions sine, cosine, the natural exponential, and the natural logarithm?

Topic 2.7 Derivatives of cos x, sin x, e to the x, and ln x: state and apply the derivatives of the four basic transcendental functions.
A focused answer to AP Calculus AB Topic 2.7, giving the derivatives of sine, cosine, the natural exponential e to the x, and the natural logarithm ln x, with worked examples combining them with the linearity rules.
8 min answer →

How are differentiability and continuity related, and where can a continuous function fail to be differentiable?

Topic 2.4 Connecting Differentiability and Continuity - Determining When Derivatives Do and Do Not Exist: explain that differentiability implies continuity but not conversely, and identify where derivatives fail to exist.
A focused answer to AP Calculus AB Topic 2.4, explaining that differentiability implies continuity but not the reverse, and identifying corners, cusps, vertical tangents and discontinuities where a derivative fails to exist.
8 min answer →

How do you estimate a derivative from a table or graph when you do not have a formula?

Topic 2.3 Estimating Derivatives of a Function at a Point: estimate the value of a derivative from a table of values or a graph using nearby secant slopes.
A focused answer to AP Calculus AB Topic 2.3, estimating a derivative numerically from a table using a symmetric difference quotient and graphically from a tangent slope, with worked examples.
8 min answer →

How does the power rule let you differentiate any power of x without going back to the limit definition?

Topic 2.5 Applying the Power Rule: differentiate power functions using the power rule, including negative and fractional exponents.
A focused answer to AP Calculus AB Topic 2.5, stating and applying the power rule for derivatives, including negative and fractional exponents after rewriting roots and reciprocals, with worked examples.
8 min answer →

Why is the derivative of a product not the product of the derivatives, and what rule replaces it?

Topic 2.8 The Product Rule: differentiate a product of two functions using the product rule.
A focused answer to AP Calculus AB Topic 2.8, stating and applying the product rule for derivatives, including products involving power, trigonometric, exponential and logarithmic factors, with worked examples.
8 min answer →

How do you differentiate a quotient of two functions, and why does the order of terms in the numerator matter?

Topic 2.9 The Quotient Rule: differentiate a quotient of two functions using the quotient rule.
A focused answer to AP Calculus AB Topic 2.9, stating and applying the quotient rule for derivatives, emphasizing the order of the numerator terms and the squared denominator, with worked examples.
8 min answer →