Chain rule

The chain rule gives the derivative of a composite function from the derivatives of its component functions. It applies recursively to nested compositions.

Composite function: A function formed by applying one function to the output of another. The chain rule differentiates this nested structure by tracking each component.
Limit of a function: The value a function approaches as its input approaches a specified point. Limits establish the derivative definition from which the chain rule follows.
Implicit differentiation: A method for finding derivatives when variables are related by an equation rather than an explicit function. Differentiating a dependent variable implicitly uses the chain rule on its hidden dependence.
Product rule: A differentiation rule stating that the derivative of a product is the first factor times the second derivative plus the reverse. It handles products of functions, while the chain rule handles nested function dependence.
Derivative: A function’s instantaneous rate of change, defined as a limit of average rates of change. The chain rule combines derivatives to obtain the rate of change of a composition.
Differentiability: The property of a function having a derivative at a point or throughout a domain. The standard chain rule requires differentiability of the component functions at the relevant points.
Related rates: A method for connecting rates of change of quantities linked by an equation. Rates in time are obtained by differentiating each quantity’s dependence on time.
Quotient rule: A differentiation rule for the quotient of two differentiable functions. It addresses division directly; nested quotients may also require the chain rule.
Leibniz notation: A notation for derivatives that writes changes in dependent variables relative to independent variables. The form dy/dx = (dy/du)(du/dx) makes intermediate-variable cancellation visible.
Linear approximation: An approximation of a function near a point by its tangent line or linear map. Composing local linear approximations explains the multiplication of derivative factors.

Linked from 34 pages