Unit 3 Study Guide: Differentiation of Composite, Implicit, and Inverse Functions (AP Calculus AB)

Composite function

A function formed by plugging one function into another, written as y = f(g(x)).

Inside function (inner function)

The function applied first in a composition, typically g(x) in f(g(x)).

Outside function (outer function)

The function applied last in a composition, typically f( ) in f(g(x)).

Chain rule

A differentiation rule for composites: if y = f(g(x)), then dy/dx = f'(g(x))·g'(x).

Leibniz form of the chain rule

Writing the chain rule as dy/dx = (dy/du)·(du/dx) when u = g(x) and y = f(u).

Function-notation chain rule

The chain rule written as d/dx[f(g(x))] = f'(g(x))·g'(x).

“Multiply the rates” interpretation

The idea that a change in x changes u = g(x), which then changes y = f(u), so rates multiply: (dy/du)(du/dx).

“Douter, inner, dinner” mnemonic

Memory trick for many chain rule problems: differentiate the outer layer, keep the inner expression, then multiply by the inner derivative.

Inner derivative factor

The g'(x) multiplier in f'(g(x))·g'(x); a common error is forgetting this factor.

Nested expression

An expression that contains another function inside it (e.g., sin(2x^2+1), (3x−1)^5), signaling chain rule.

Product rule

Derivative rule for a product: d/dx[uv] = u'v + uv'.

Quotient rule

Derivative rule for a quotient: d/dx[u/v] = (u'v − uv')/v^2.

Product of composites

An expression like (x^2+1)^3(5x−7)^4 where you use product rule first, then chain rule inside each factor.

Quotient with a composite numerator/denominator

An expression like sin(3x)/(x^2+1) where you use quotient rule, and chain rule when differentiating sin(3x).

Derivative of e^x

d/dx[e^x] = e^x.

Derivative of a^x

d/dx[a^x] = a^x ln(a).

Derivative of ln(x)

d/dx[ln(x)] = 1/x (for x>0).

Derivative of e^{g(x)}

Chain-rule pattern: d/dx[e^{g(x)}] = e^{g(x)}·g'(x).

Derivative of ln(g(x))

Chain-rule pattern: d/dx[ln(g(x))] = g'(x)/g(x) (“inner derivative over the inner”).

Logarithm domain restriction

For ln(g(x)) to be defined (and differentiable), the argument must be positive: g(x) > 0.

Implicit equation

An equation where y is not isolated (e.g., x^2 + y^2 = 25), representing a relation instead of an explicit y=f(x).

Implicit differentiation

Differentiating both sides with respect to x while treating y as a function of x, so y-terms produce dy/dx factors.

Treat y as a function of x

Key implicit idea: y depends on x, so dy/dx is not 0 and must appear when differentiating y-expressions.

Derivative of y with respect to x

In implicit work, d/dx[y] = dy/dx (not 0).

Derivative of y^n (implicit)

Using chain rule: d/dx[y^n] = n·y^(n−1)·(dy/dx).

dy/dx notation

A derivative notation emphasizing that y depends on x; important for chain rule and implicit differentiation.

Reciprocal derivatives (dx/dy and dy/dx)

When defined and nonzero, dx/dy = 1/(dy/dx); useful for checking or converting results.

Tangent line slope

At a point on a curve, the tangent slope is m_tan = dy/dx evaluated at that point.

Point-slope form

A line form used for tangent/normal lines: y − y1 = m(x − x1).

Normal line slope

Slope of the line perpendicular to the tangent at a point: mnorm = −1/mtan (when m_tan ≠ 0).

Negative reciprocal

The relationship between perpendicular slopes: if a line has slope m, a perpendicular line has slope −1/m.

Vertical tangent

A tangent line with undefined slope; in that case, the normal line is horizontal.

Horizontal tangent

A tangent line with slope 0; in that case, the normal line is vertical.

Second derivative (implicit)

d^2y/dx^2 found by differentiating dy/dx with respect to x while remembering y still depends on x.

Concavity

The curve’s bending behavior determined by the sign of d^2y/dx^2 (negative means concave down, positive means concave up).

Quotient rule in implicit second derivatives

Often needed when dy/dx is a quotient involving y (e.g., dy/dx = −x/y), since y depends on x.

Simplify before taking a second derivative

A practical strategy: algebraically simplify dy/dx first to reduce errors when differentiating again.

Inverse function

A function that reverses input-output: if f(a)=b then f^{-1}(b)=a.

One-to-one function

A function where each output corresponds to exactly one input; required for an inverse to be a function (often ensured by being strictly increasing/decreasing).

Reflection across y = x

Graph relationship: the graph of f^{-1} is the reflection of the graph of f across the line y=x.

Inverse derivative formula (general)

If f and f^{-1} are inverses, then (f^{-1})'(x) = 1 / f'(f^{-1}(x)), assuming the denominator is nonzero.

Inverse derivative formula (point-specific)

If f(a)=b, then (f^{-1})'(b) = 1/f'(a); points swap between f and f^{-1}.

Corresponding points swap

If (a,b) lies on f, then (b,a) lies on f^{-1}; use the swapped point to evaluate inverse derivatives.

f^{-1}(x) vs 1/f(x)

f^{-1}(x) means the inverse function, not the reciprocal; confusing these is a common mistake.

Inverse trigonometric function

A function like arcsin, arccos, or arctan that undoes a trig function on a restricted domain so it is one-to-one.

Principal range

The restricted output interval chosen to make an inverse trig function a true function (single output for each input).

Derivative of arcsin(x)

d/dx[arcsin(x)] = 1/√(1−x^2) (defined for −1<x<1; denominator goes to 0 at ±1).

Derivative of arccos(x)

d/dx[arccos(x)] = −1/√(1−x^2) (same domain considerations as arcsin).

Derivative of arctan(x)

d/dx[arctan(x)] = 1/(1+x^2).

Chain rule with inverse trig

Patterns: d/dx[arcsin(g(x))] = g'(x)/√(1−(g(x))^2) and d/dx[arctan(g(x))] = g'(x)/(1+(g(x))^2).