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AP Calculus AB/BC Unit 3 Visual ReviewA topic-by-topic visual walkthrough of Unit 3: Composite, Implicit & Inverse Functions — the chain rule, implicit differentiation, inverse derivatives, and higher-order derivatives.
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TOPIC 3.1
The Chain Rule
d/dx[ f(g(x)) ] = f′(g(x)) · g′(x)
Outside, then inside
y = (3x² + 1)⁵
outer = u⁵, inner = 3x²+1
y′ = 5(3x²+1)⁴ · 6x
= 30x(3x²+1)⁴
Differentiate the outer, keep inner, ×inner′.
More examples
d/dx[sin(2x)] = cos(2x)·2
d/dx[e^(x²)] = e^(x²)·2x
d/dx[ln(5x)] = (1/5x)·5 = 1/x
Chain outward through each layer, one at
a time, for deeply nested functions.
Chain rule: (f∘g)′ = f′(g(x))·g′(x) — outer derivative times inner derivative.The Review Hub · AP Calculus AB/BC Unit 3
TOPIC 3.2
Implicit Differentiation
Differentiate both sides in x
x² + y² = 25
2x + 2y·(dy/dx) = 0
dy/dx = −x / y
Each y-term gets a dy/dx factor (chain
rule), since y is a function of x.
The steps
1. d/dx BOTH sides (treat y = y(x)).
2. Every y differentiates to y′·dy/dx.
3. Collect the dy/dx terms.
4. Solve for dy/dx.
Answer may contain both x and y.
Use when y can't be isolated
Essential for curves like circles, ellipses, and other relations not solved for y.
Differentiate both sides , attach dy/dx to each y, then solve for dy/dx.The Review Hub · AP Calculus AB/BC Unit 3
TOPIC 3.3
Differentiating Inverse Functions
(f⁻¹)′(x) = 1 / f′( f⁻¹(x) )
Reciprocal slope at matched points
If f(a) = b, then f⁻¹(b) = a, and
(f⁻¹)′(b) = 1 / f′(a)
The tangent slopes of f and f⁻¹ at
corresponding points are RECIPROCALS.
Because their graphs reflect over y = x.
Worked example
f(x)=x³+x, f(1)=2, f′=3x²+1
Find (f⁻¹)′(2):
= 1 / f′(1) = 1 / 4
You need the x that maps to the input —
here f(1) = 2, so use f′(1).
↔(f⁻¹)′(b) = 1/f′(a) where f(a)=b — inverse slopes are reciprocals.
The Review Hub · AP Calculus AB/BC Unit 3
TOPIC 3.4
Inverse Trig Derivatives
// on the AP formula sheet — recognize them
d/dx[arcsin x] = 1 / √(1 − x²)
d/dx[arctan x] = 1 / (1 + x²)
d/dx[arcsec x] = 1 / (|x|√(x² − 1))
Combine with the chain rule
d/dx[arctan(3x)] = 1/(1+(3x)²) · 3 = 3/(1+9x²). The "co-" inverses (arccos, arccot) are just negatives.
(arcsin)′=1/√(1−x²), (arctan)′=1/(1+x²) — chain-rule the inner function.The Review Hub · AP Calculus AB/BC Unit 3
TOPIC 3.5
Selecting Procedures for Derivatives
Identify the STRUCTURE first
• Product of factors → product rule
• A fraction → quotient rule
• A composition (nesting) → chain rule
• Just powers/sums → power & sum rules
Often you'll combine several rules.
Simplify before differentiating
Algebra can turn a hard derivative into
an easy one:
(x²+x)/x = x + 1 → deriv 1
Expand products, split fractions, or use
log properties before applying rules.
Nested rules
e.g. x²·sin(3x) needs the product rule AND, on the second factor, the chain rule. Work from the outside in.
Read the structure to pick the rule(s); simplify with algebra when you can.The Review Hub · AP Calculus AB/BC Unit 3
TOPIC 3.6
Higher-Order Derivatives
f′ → f″ → f‴ → f⁽ⁿ⁾ (differentiate repeatedly)
Worked example
f(x) = x⁴
f′ = 4x³
f″ = 12x²
f‴ = 24x, f⁗ = 24
Notation: f″(x) = d²y/dx²
What they mean
f′ = rate of change (slope / velocity).
f″ = rate of change of the rate
(concavity / acceleration).
The second derivative drives concavity
and inflection analysis in Unit 5.
Differentiate repeatedly: f″ is the rate of change of f′ (concavity, acceleration).
The Review Hub · AP Calculus AB/BC Unit 3
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How to use the visual review
Spend 30 seconds per slide before clicking next. Look at the diagram, then ask yourself: "Could I state this rule, or differentiate this function, from memory?"
Use the fullscreen button () on desktop for the best experience. Use arrow keys to navigate. Tap "Show all slides" to jump around.
This is great for review the night before the exam — fast, visual, and covers every idea you need to recognize in Unit 3.