AP Calculus BC Unit 3 Study Guide: Composite, Implicit, Logarithmic, and Inverse Differentiation

Composite function

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

Inside function

In a composite f(g(x)), g(x) is the inner function whose output becomes the input to the outer function.

Outside function

In a composite f(g(x)), f is the outer function that acts on the result of the inner function.

Function composition notation

The notation f(g(x)) meaning “apply g to x, then apply f to that result.”

Two-step machine interpretation

A way to view f(g(x)) as: x → g(x) → f(g(x)), emphasizing layered change.

Chain Rule (standard form)

If y=f(g(x)), then dy/dx = f'(g(x))·g'(x).

Chain Rule (Leibniz/substitution form)

If y=y(v) and v=v(x), then dy/dx = (dy/dv)·(dv/dx).

Chained rates of change idea

The reason the chain rule multiplies: overall change = (outer rate wrt inner) × (inner rate wrt x).

“douter, inner, dinner”

Memory aid for chain rule: differentiate the outer, keep the inner, then multiply by the derivative of the inner.

Power chain rule pattern

If u=u(x), then d/dx(u^n)=n·u^(n−1)·u'.

Chain rule trigger

You need the chain rule when a basic function is applied to an input that is not just x (i.e., a nontrivial inner expression).

Nontrivial inner expression

An inner input like 3x^2−5x+1, √(x+2), or x^3 that requires differentiating as u(x), not as x.

Composition without parentheses

Composition can be implied, e.g., sin x^2 usually means sin(x^2), even without parentheses.

Common chain rule mistake: missing inner derivative

An error where you differentiate the outer function but forget to multiply by g'(x).

Radical as a power

Rewriting √(x+2) as (x+2)^(1/2) to apply power and chain rules cleanly.

Negative exponent form

A way to rewrite a denominator, e.g., 1/√(x+2) as (x+2)^(−1/2), to differentiate using the power rule.

Exponential chain rule (base e)

If u=u(x), then d/dx(e^u)=e^u·u'.

Exponential chain rule (base a)

If u=u(x), then d/dx(a^u)=a^u·ln(a)·u'.

Logarithm chain rule

If u=u(x), then d/dx(ln u)=u'/u.

Sine chain rule

If u=u(x), then d/dx(sin u)=cos(u)·u'.

Cosine chain rule

If u=u(x), then d/dx(cos u)=−sin(u)·u'.

Tangent chain rule

If u=u(x), then d/dx(tan u)=sec^2(u)·u'.

Silent mistake in ln differentiation

Writing d/dx(ln u)=1/u instead of (u'/u), forgetting the chain rule factor u'.

Product Rule

For y=p(x)q(x), y' = p'(x)q(x) + p(x)q'(x).

Product + Chain Rule combination

Use product rule for the outer multiplication, then apply chain rule inside any composite factor derivatives.

Common product+chain mistake (e^{3x})

Differentiating e^{3x} as e^{3x} instead of 3e^{3x} (missing the inner derivative 3).

Quotient Rule

For y=p(x)/q(x), y' = (p'(x)q(x) − p(x)q'(x)) / (q(x))^2.

Quotient + Chain Rule combination

Use quotient rule for the overall fraction, then chain rule inside p'(x) and/or q'(x) if they are composite.

Common quotient+chain mistake (sin(x^2))

Differentiating sin(x^2) as cos(x^2) and forgetting to multiply by 2x.

Repeated (multi-layer) chain rule

Applying the chain rule multiple times for nested layers, working from the outermost layer inward.

Structure recognition (outermost operation)

A strategy: identify whether the outermost structure is sum, product, quotient, or composition before differentiating.

Explicit equation

An equation where y is isolated as a function of x (e.g., y=x^2+3).

Implicit equation

A relationship mixing x and y (e.g., x^2+y^2=25) where y may not be a single function of x globally.

Implicit differentiation

Differentiating both sides with respect to x while treating y as y(x), so derivatives of y-terms include dy/dx.

Chain rule for y-terms (implicit)

When differentiating a term involving y, multiply by dy/dx because y depends on x.

Core implicit differentiation procedure

Differentiate both sides, include dy/dx on y-terms, collect dy/dx terms, factor dy/dx, then solve for dy/dx.

Implicit differentiation mistake: d/dx(y^2)

Incorrectly writing d/dx(y^2)=2y instead of 2y·(dy/dx).

Product rule in implicit problems (xy)

For xy where both x and y vary with x, d/dx(xy)=x(dy/dx)+y.

Tangent line slope (implicit curve)

The slope of the tangent line at (x0,y0) is dy/dx evaluated at that point.

Point-slope form

A line form y−y0 = m(x−x0), often used after finding a tangent slope at a point.

Normal line slope

The slope perpendicular to the tangent: mnormal = −1/mtangent (when m_tangent ≠ 0).

Second derivative via implicit differentiation

After finding dy/dx implicitly, differentiate that expression again with respect to x, still treating y as dependent on x.

Reciprocal derivative relationship

When derivatives exist and dy/dx ≠ 0, dx/dy = 1/(dy/dx).

Logarithmic differentiation

A method that takes ln of both sides to simplify differentiating products/quotients/powers, especially with variable exponents.

Log property: product

ln(ab)=ln(a)+ln(b), used to turn multiplication into addition.

Log property: quotient

ln(a/b)=ln(a)−ln(b), used to turn division into subtraction.

Log property: power

ln(a^k)=k·ln(a), used to bring exponents down as multipliers.

Derivative of x^x (via logs)

d/dx(x^x)=x^x(ln x + 1), found by ln(y)=x ln x then differentiating.

Inverse functions

Functions f and f^{-1} that undo each other: f(f^{-1}(x))=x and f^{-1}(f(x))=x.

Derivative of an inverse function (point form)

If b=f(a) and f'(a)≠0, then (f^{-1})'(b)=1/f'(a).