Rucete ✏ AP Calculus In a Nutshell
3. Differentiation
This chapter defines the derivative, reviews rules for finding derivatives of various functions, introduces important theorems like the Mean Value Theorem, and covers advanced topics including implicit differentiation and L'Hospital's Rule.
Definition of Derivative
• The derivative of f at x = a is:
f′(a) = limh→0 [f(a + h) – f(a)] / h
or
f′(a) = limx→a [f(x) – f(a)] / (x – a)
• It represents the instantaneous rate of change or slope of the tangent line at x = a.
Second Derivative
• Denoted as f″(x), it is the derivative of the derivative → tells concavity.
Basic Derivative Formulas
Let a and n be constants, and u and v differentiable functions of x:
• d/dx (a) = 0
• d/dx (xⁿ) = n·xⁿ⁻¹
• d/dx (a·f(x)) = a·f′(x)
• d/dx (f ± g) = f′ ± g′
• Product Rule: d/dx (uv) = u′v + uv′
• Quotient Rule: d/dx (u/v) = (u′v – uv′)/v²
Chain Rule
• If y = f(g(x)), then y′ = f′(g(x)) · g′(x)
• “Differentiate the outside, multiply by derivative of inside”
Example: If y = (3x + 2)⁵, then y′ = 5(3x + 2)⁴ · 3 = 15(3x + 2)⁴
Differentiability and Continuity
• If f is differentiable at x = c, then f is continuous at x = c.
• But continuity does not imply differentiability.
• Non-differentiable cases: – Corner (e.g., |x| at x = 0) – Cusp – Vertical tangent – Discontinuity
Estimating Derivatives
Numerically:
• Use symmetric difference quotient:
f′(a) ≈ [f(a + h) – f(a – h)] / (2h)
Graphically:
• Estimate slope of tangent line to f(x) at several points to sketch f′(x).
Differentiation of Parametric Equations (BC)
• Given x = f(t), y = g(t):
– dy/dx = (dy/dt) / (dx/dt)
– d²y/dx² = [d/dt(dy/dx)] / (dx/dt)
Example: x = t², y = t³ → dy/dx = 3t² / 2t = (3/2)t
Implicit Differentiation
• Use when y is not isolated
• Differentiate both sides of the equation with respect to x – Treat y as a function of x – Apply chain rule to terms with y (dy/dx appears)
Example: x² + y² = 25 → 2x + 2y(dy/dx) = 0 → dy/dx = –x/y
Derivative of Inverse Functions
• If f and f⁻¹ are inverse functions: (f⁻¹)'(a) = 1 / f'(f⁻¹(a))
Example: If f(2) = 5 and f′(2) = 3 → f⁻¹(5) = 2, so (f⁻¹)'(5) = 1/3
Derivatives of Inverse Trig Functions
• d/dx [sin⁻¹(x)] = 1 / √(1 – x²)
• d/dx [cos⁻¹(x)] = –1 / √(1 – x²)
• d/dx [tan⁻¹(x)] = 1 / (1 + x²)
Mean Value Theorem (MVT)
• If f is continuous on [a, b] and differentiable on (a, b), then ∃ c ∈ (a, b) such that:
f′(c) = [f(b) – f(a)] / (b – a)
• Guarantees that some tangent line is parallel to secant line over [a, b]
Rolle’s Theorem
• Special case of MVT where f(a) = f(b)
• Then ∃ c ∈ (a, b) where f′(c) = 0 (horizontal tangent)
Example Graph: f(x) = (x – 1)(x – 3) on [1, 3]
L'Hospital's Rule
• If lim f(x)/g(x) = 0/0 or ∞/∞, then:
lim f(x)/g(x) = lim f′(x)/g′(x), if limit on right exists
Example: limx→0 sin(x)/x = limx→0 cos(x)/1 = 1
Recognizing Derivatives in Limits
• Some limits are derivatives in disguise:
limh→0 [f(a + h) – f(a)] / h = f′(a)
limx→a [f(x) – f(a)] / (x – a) = f′(a)
In a Nutshell
Differentiation captures how functions change. You can apply the derivative to algebraic, parametric (BC), or implicitly defined functions. Derivatives have geometric meaning (slopes) and analytic applications (rates, MVT, L'Hospital). Mastering rules and interpretation is key to solving a wide range of calculus problems.