AP Calc AB · Unit 2 of 8

Differentiation: Definition and Basic Rules

Build the derivative from first principles, interpret derivative values, and use core differentiation rules for common function forms.

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Why this unit matters

The derivative is the main tool for describing instantaneous rate of change and local linear behavior.

What you will learn

  • Compute derivatives from the limit definition in simple cases.
  • Interpret derivative values and units in context.
  • Differentiate polynomial, power, exponential, and trigonometric functions using basic rules.

Understand the core ideas

The derivative f'(x) measures instantaneous rate of change, which you can think of as the slope of the tangent line at x. AP Calculus AB starts with the limit definition because it explains where derivative rules come from: f'(x) = lim h->0 (f(x + h) - f(x)) / h. That quotient is an average rate over a tiny interval of width h. As h shrinks toward 0, the average rate approaches the instantaneous rate when the derivative exists.

After the definition, derivative rules let you work efficiently and focus on interpretation. The power rule, constant multiple rule, and sum or difference rule combine to handle many algebraic models quickly. You also need standard derivatives for exe^x, sin x, and cos x in AB problems. Keep units in mind: if s(t) is meters, then s'(t) is meters per second. A positive derivative means increasing output as input increases, while a negative derivative means decreasing output. A zero derivative can indicate a horizontal tangent and possible local turning behavior.

Key terms

Derivative
The instantaneous rate of change of a function with respect to its input, represented by f'(x) or dy/dx.
Difference quotient
The expression (f(x + h) - f(x)) / h, which gives average rate of change over an interval of width h.
Tangent line
A line that touches a curve at a point and has slope equal to the derivative at that point.
Power rule
For n a constant, d/dx (xn)=n(x^n) = n*x(n−1)x^(n-1).

Find a tangent line using derivative rules

For f(x)=x3−4xf(x) = x^3 - 4x, find the tangent line at x = 2.

  1. Differentiate: f′(x)=3x2−4f'(x) = 3x^2 - 4.
  2. Evaluate slope at x = 2: f'(2) = 3(4) - 4 = 8.
  3. Find the point on the curve: f(2) = 8 - 8 = 0, so the point is (2, 0).
  4. Use point-slope form: y - 0 = 8(x - 2), then simplify to y = 8x - 16.
Result: The tangent line is y = 8x - 16.

A common misconception

Claim: If f'(a) = 0, then a is always a local maximum or minimum.

Correction: f'(a) = 0 marks a critical point candidate. You still need sign analysis or another test, since it could be neither max nor min.

Lessons in this unit

  1. Derivative as a limit of difference quotientsMove from average rate of change to instantaneous rate of change. Read the full guide →
  2. Notation and tangent line meaningUse f'(x), dy/dx, and derivative values to describe local behavior.
  3. Power, sum, and constant multiple rulesDifferentiate efficiently without rebuilding limits each time. Read the full guide →
  4. Derivatives of trig and exponential functionsApply standard derivatives and combine them in mixed expressions.

Study task

For s(t)=t3−6t2+9ts(t) = t^3 - 6t^2 + 9t, compute the average rate of change on [1, 3] and the instantaneous rate at t = 2, then compare meanings.

Unit checkpoint

Using derivative rules, find d/dx of 3x4−5x2+73x^4 - 5x^2 + 7.

The derivative is 12x3−10x12x^3 - 10x.

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