A composite function f(g(x)) nests one function inside another. The chain rule differentiates it by working outside-in: d/dx[f(g(x))] = f′(g(x))·g′(x) — the outer derivative (evaluated at the inner function) times the derivative of the inner function. This single rule powers most of Unit 3, giving shortcuts like d/dx[sin(u)] = cos(u)·u′ and d/dx[uⁿ] = n·uⁿ⁻¹·u′.
Chain RuleComposite FunctionsOutside-In
Key Concept 2
Implicit differentiation handles equations not solved for y
When an equation like x² + y² = 25 isn't solved for y, use implicit differentiation: differentiate both sides with respect to x, treating y as a function of x so every y-term picks up a factor of dy/dx (the chain rule at work), then solve algebraically for dy/dx. This same chain-rule idea gives the inverse function derivative, (f⁻¹)′(x) = 1/f′(f⁻¹(x)) — the slope of an inverse is the reciprocal of the original's slope at the matching point.
Implicitdy/dxInverse Derivative
Key Concept 3
Inverse trig and higher-order derivatives round out the toolkit
The inverse trigonometric functions have memorized derivatives: arcsin x → 1/√(1−x²), arccos x → −1/√(1−x²), and arctan x → 1/(1+x²). Topic 3.5 is about selecting the right procedure for a given function, and Topic 3.6 covers higher-order derivatives — found by differentiating repeatedly, where the second derivative measures acceleration and concavity.
Inverse TrigSelecting ProceduresHigher-Order
Composite function
f(g(x)); the output of g is the input of f.
Chain Rule
Chain Rule
d/dx[f(g(x))] = f′(g(x))·g′(x).
Chain Rule
Inner / outer function
In f(g(x)), g is the inner function and f is the outer function.
Chain Rule
Chain rule for uⁿ
d/dx[uⁿ] = n·uⁿ⁻¹·u′.
Chain Rule
Chain rule for sin/e/ln
cos(u)·u′, e^u·u′, and u′/u.
Chain Rule
Implicit differentiation
Differentiating both sides w.r.t. x with y as a function of x.
Implicit
dy/dx factor
The chain-rule factor attached to each y-term when differentiating implicitly.
Implicit
Solving for dy/dx
Algebraically isolating dy/dx after implicit differentiation.
Implicit
Inverse function
f⁻¹ undoes f; its graph is the reflection of f over y = x.
Inverse
Inverse function derivative
(f⁻¹)′(x) = 1/f′(f⁻¹(x)).
Inverse
arcsin derivative
d/dx[arcsin x] = 1/√(1 − x²).
Inverse Trig
arccos derivative
d/dx[arccos x] = −1/√(1 − x²).
Inverse Trig
arctan derivative
d/dx[arctan x] = 1/(1 + x²).
Inverse Trig
Selecting procedures
Choosing chain, product, quotient, or implicit methods for a function.
Procedures
Higher-order derivative
The derivative of a derivative (f″, f‴, …).
Higher-Order
Second derivative
f″ = d²y/dx²; the rate of change of f′.
Higher-Order
Concavity
The direction a curve bends, governed by the sign of f″.