Differentiation: Definition — AP Calculus AB Unit 2 practice.
This unit covers derivative definition, basic rules and tangent lines — essential concepts for AP Calculus AB. Use our interactive study games to test your understanding, or review questions in traditional format below.
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This unit covers derivative definition, basic rules and tangent lines — essential concepts for AP Calculus AB. Use our interactive study games to test your understanding, or review questions in traditional format below.
Key Concepts Breakdown
1 Derivative Definition
The derivative of a function f at a point x is defined as the limit of the difference quotient: f'(x) = lim(h→0) [f(x+h) - f(x)] / h. Students must be able to apply this definition directly to find derivatives and recognize when a limit represents a derivative. The exam may present the limit in alternate forms, such as lim(x→a) [f(x) - f(a)] / (x - a), and expect students to identify what it computes.
Key Points
- f'(a) = lim(h→0) [f(a+h) - f(a)] / h — memorize this form exactly
- If this limit does not exist, the function is not differentiable at that point
- Differentiability implies continuity; continuity does NOT imply differentiability
- A function fails to be differentiable at corners, cusps, vertical tangents, and discontinuities
Let f(x) = x². Use the limit definition to find f'(3).
Set up the difference quotient: lim(h→0) [(3+h)² - 9] / h = lim(h→0) [9 + 6h + h² - 9] / h = lim(h→0) [6h + h²] / h. Factor h from the numerator to get lim(h→0) (6 + h), which equals 6. So f'(3) = 6.
2 Basic Differentiation Rules
Students must fluently apply the power rule, constant rule, constant multiple rule, and sum/difference rule without reaching for the limit definition. The exam expects instant recall of these rules to differentiate polynomial, radical, and simple rational functions written as power functions. Speed and accuracy with these rules is essential since they underlie every subsequent differentiation problem.
Key Points
- Power Rule: d/dx [xⁿ] = nxⁿ⁻¹ for any real n — applies to negative and fractional exponents too
- Constant Rule: d/dx [c] = 0
- Constant Multiple Rule: d/dx [c·f(x)] = c·f'(x)
- Sum/Difference Rule: d/dx [f(x) ± g(x)] = f'(x) ± g'(x)
Find dy/dx for y = 4x³ - 7x + (3/x²) + √x.
Rewrite as y = 4x³ - 7x + 3x⁻² + x^(1/2) so the power rule applies to every term. Differentiating term by term: dy/dx = 12x² - 7 + 3(-2)x⁻³ + (1/2)x^(-1/2). Simplifying: dy/dx = 12x² - 7 - 6/x³ + 1/(2√x). The key step is rewriting radicals and fractions as power functions before differentiating.
3 Tangent Lines
The derivative f'(a) gives the slope of the tangent line to the graph of f at x = a. Students must be able to write the equation of a tangent line using point-slope form and, separately, identify the equation of a normal line (perpendicular to the tangent). The exam frequently asks for tangent lines at a given point or for values of x where the tangent line has a specified slope.
Key Points
- Tangent line at x = a: y - f(a) = f'(a)(x - a) — always use point-slope form
- The slope of the normal line is -1/f'(a), the negative reciprocal of the tangent slope
- To find where the tangent is horizontal, set f'(x) = 0; for vertical, look for where f'(x) is undefined
- The tangent line touches the curve at exactly one point locally but may cross the curve elsewhere
Find the equation of the line tangent to f(x) = x³ - 2x at x = 1.
First find the y-coordinate of the point: f(1) = 1 - 2 = -1, giving the point (1, -1). Next differentiate to get f'(x) = 3x² - 2, then evaluate f'(1) = 3(1)² - 2 = 1, so the tangent slope is 1. Applying point-slope form: y - (-1) = 1(x - 1), which simplifies to y = x - 2.
Questions, answered.
What is Differentiation: Definition?
Differentiation: Definition is Unit 2 of AP Calculus AB, covering derivative definition, basic rules and tangent lines.
How to study for AP Calculus AB Unit 2?
Start with the Quick Summary above, review the Key Concepts, then test yourself with our interactive study games. Aim for 80%+ accuracy before moving on.
How many questions are in this unit?
This unit has 30+ review questions across 5 different game modes.