Unit 2: Differentiation: Definition & Properties

📚Study Guide: Differentiation: Definition & Properties

Unit 2: Differentiation: Definition and Properties

This unit introduces the derivative, one of the two central pillars of calculus. The derivative measures the instantaneous rate of change of a function at a point and is defined as the limit of the difference quotient. You will learn to interpret derivatives as slopes of tangent lines, velocities, and rates of change. After establishing the limit definition, you will develop differentiation rules that allow you to compute derivatives efficiently: the power rule, constant rule, constant multiple rule, sum and difference rules, and the product and quotient rules. You will also explore higher-order derivatives and their notations. Understanding the conceptual foundation here is critical because every subsequent unit builds upon these computational skills.

Key Concepts

• Derivative as Limit: f'(x) = lim(h->0) [f(x+h) - f(x)]/h. This is the slope of the tangent line at x.
• Alternate Form: f'(c) = lim(x->c) [f(x) - f(c)]/(x - c). Useful when analyzing specific points.
• Differentiability Implies Continuity: If f is differentiable at c, it must be continuous at c. The converse is false (e.g., |x| at 0).
• Power Rule: d/dx [x^n] = n*x^(n-1) for all real numbers n.
• Product Rule: d/dx [uv] = u'v + uv'. Do not simply multiply derivatives.
• Quotient Rule: d/dx [u/v] = (u'v - uv')/v^2. Remember the minus and squared denominator.
• Higher-Order Derivatives: f'', f''', etc., represent acceleration and higher rates of change.
• Tangent Line Equation: y - f(a) = f'(a)(x - a). You need the point and the slope.

Vocabulary

• Derivative: The instantaneous rate of change of a function with respect to its input variable.
• Difference quotient: The expression [f(x+h) - f(x)]/h representing average rate of change over interval h.
• Tangent line: A line that touches a curve at exactly one point locally and has slope equal to the derivative there.
• Secant line: A line passing through two points on a curve; its slope is the average rate of change.
• Differentiable: A function is differentiable at a point if its derivative exists there.
• Normal line: A line perpendicular to the tangent line at a point of tangency.

Formulas

• f'(x) = lim(h->0) [f(x+h) - f(x)]/h
• d/dx [x^n] = n*x^(n-1)
• d/dx [cf(x)] = c*f'(x)
• d/dx [f(x) + g(x)] = f'(x) + g'(x)
• d/dx [f(x)g(x)] = f'(x)g(x) + f(x)g'(x)
• d/dx [f(x)/g(x)] = [f'(x)g(x) - f(x)g'(x)] / [g(x)]^2
• Tangent line: y = f(a) + f'(a)(x - a)

Common Mistakes

• Applying the power rule to exponential functions: d/dx [2^x] is NOT x*2^(x-1); use ln(2)*2^x.
• Forgetting to square the denominator in the quotient rule.
• Assuming differentiability implies all derivatives exist everywhere; functions can be differentiable but not twice differentiable.
• Using the product rule when one factor is a constant; the constant multiple rule is simpler and less error-prone.

AP Exam Strategies

• When an FRQ asks for a derivative using the limit definition, write the full difference quotient setup before simplifying.
• On multiple choice, if you forget the quotient rule, rewrite the quotient as u*v^(-1) and use the product rule.
• Always check whether a piecewise function is differentiable at the boundary by confirming continuity and equal left/right derivatives.
• For tangent line problems, verify that your line actually passes through the given point as a quick sanity check.

Real-World Applications

• Physics: Velocity is the derivative of position; acceleration is the derivative of velocity.
• Economics: Marginal cost is the derivative of the cost function, approximating the cost of producing one additional unit.
• Medicine: The rate of drug concentration in the bloodstream is modeled by derivatives to determine safe dosing intervals.