Differentiating Composites and Implicit Relations (AP Calculus BC Unit 3 Notes)

Composite function

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

Chain rule

A differentiation rule for composite functions: d/dx[f(g(x))] = f'(g(x))·g'(x).

Outer function

In a composite function f(g(x)), the “outside” operation f( ) applied last (e.g., sin( ) in sin(x^2)).

Inner function

In a composite function f(g(x)), the “inside” input g(x) that gets fed into the outer function (e.g., x^2 in sin(x^2)).

Layer (in chain rule)

One step/operation in a multi-step composite function; differentiating requires multiplying derivatives from each layer.

Leibniz chain rule form

A chain rule notation that shows intermediate variables: dy/dx = (dy/du)·(du/dx), suggesting the “u” cancels conceptually.

Derivative (rate of change)

A measure of how fast an output changes with respect to an input; interpreted as the instantaneous rate of change or slope.

Operator notation

A derivative written as an operator acting on a function, such as (d/dx)[f(x)].

Prime notation

A derivative written with primes, such as y' or f'(x).

Common chain rule error

Differentiating the outer function but forgetting to multiply by the derivative of the inner function (the “inside” factor).

Parentheses/structure error

A mistake where missing or misplaced parentheses changes meaning (e.g., cos((2x−1)^4) vs. cos(2x−1)^4).

Implicit differentiation

A method for finding dy/dx when x and y are mixed together by differentiating both sides with respect to x and treating y as y(x).

Explicit equation/function

A relationship where y is isolated as a formula in terms of x (e.g., y = x^2 + 3).

Implicit equation

A relationship where x and y appear together and y is not solved for directly (e.g., x^2 + y^2 = 25).

Dependent variable idea in implicit differentiation

The principle that y depends on x (y = y(x)), so differentiating y-terms requires chain rule factors of dy/dx.

Signature of implicit differentiation

The appearance of dy/dx when differentiating expressions containing y, e.g., d/dx(y^2) = 2y·dy/dx.

Implicit differentiation workflow

Differentiate both sides, attach dy/dx to y-terms, collect dy/dx terms on one side, factor dy/dx, then solve for dy/dx.

Circle slope (implicit)

For x^2 + y^2 = 25, implicit differentiation gives dy/dx = −x/y.

Vertical tangent (implicit curve)

A point where the slope is undefined; for dy/dx = −x/y on a circle, this occurs when y = 0.

Tangent line (implicit curve)

A line found by computing dy/dx, evaluating the slope at a point (x1,y1), then using point-slope form.

Point-slope form

An equation of a line through (x1,y1) with slope m: y − y1 = m(x − x1).

Higher-order derivative

A derivative taken more than once (the derivative of a derivative), such as the second derivative or third derivative.

Second derivative

The derivative of the first derivative; written y'' or d^2y/dx^2, measuring how the slope/rate of change is changing.

Concavity (via second derivative)

A graph is concave up where y'' > 0 and concave down where y'' < 0 (for smooth explicit functions).

Second derivative for the circle

Starting from dy/dx = −x/y (from x^2+y^2=25), differentiating again gives d^2y/dx^2 = −(x^2+y^2)/y^3 = −25/y^3 on the circle.