Skip to main content

Mathematics for Physics

306 Chain Rule

A differentiation rule for a composition of functions that is used to compute the derivative of a composite mapping from the derivatives of its inner and outer factors.

Note: Also called the composite function rule.

definition [d] (Chain Rule) From Stewart: if \(g\) is differentiable at \(x\) and \(f\) is differentiable at \(g(x)\), then the composite \(F = f \circ g\) defined by \(F(x) = f(g(x))\) is differentiable at \(x\) and

  • \(\dfrac{dF}{dx} = \dfrac{df}{du}\bigg|_{u=g(x)}\, \dfrac{dg}{dx}\) .

Equivalently, if \(y = f(u)\) and \(u = g(x)\), then

  • \(\dfrac{dy}{dx} = \dfrac{dy}{du}\, \dfrac{du}{dx}\) .

where

  • \(g\) is the inner function.
  • \(f\) is the outer function.
  • \(F = f \circ g\) is the composite function.
  • \(x\) is the independent variable.
  • \(u = g(x)\) is the intermediate variable.
  • \(y = f(u)\) is the dependent variable.
  • \(\dfrac{dF}{dx}\) is the derivative of \(F\) with respect to \(x\).
  • \(\dfrac{df}{du}\bigg|_{u=g(x)}\) is the derivative of \(f\) with respect to \(u\), evaluated at \(u = g(x)\).
  • \(\dfrac{dg}{dx}\) is the derivative of \(g\) with respect to \(x\).
  • \(\dfrac{dy}{dx}\) is the derivative of \(y\) with respect to \(x\).
  • \(\dfrac{dy}{du}\) is the derivative of \(y\) with respect to \(u\).
  • \(\dfrac{du}{dx}\) is the derivative of \(u\) with respect to \(x\).

definition [d] (Chain Rule) From Stewart: if \(n\) is any real number and \(u = g(x)\) is differentiable, then

  • \(\dfrac{d}{dx}\bigl[u^{n}\bigr] = n u^{n-1}\, \dfrac{du}{dx}\) .

where

  • \(n\) is a real exponent.
  • \(u = g(x)\) is a differentiable function of \(x\).
  • \(x\) is the independent variable.
  • \(\dfrac{d}{dx}\) is differentiation with respect to \(x\).
  • \(\dfrac{du}{dx}\) is the derivative of \(u\) with respect to \(x\).

306.1 Examples

306.1.1 Simple

For \(y = (x^{2} + 1)^{3}\), take \(u = x^{2} + 1\) and \(y = u^{3}\).

\[ \dfrac{dy}{dx} = 3u^{2}\, \dfrac{du}{dx} = 3(x^{2} + 1)^{2}\cdot (2x) = 6x(x^{2} + 1)^{2} \]

where

  • \(x\) is the independent variable.
  • \(u = x^{2} + 1\) is the inner function.
  • \(y = u^{3}\) is the outer function of \(u\).
  • \(\dfrac{du}{dx} = 2x\) is the derivative of the inner function.
  • \(\dfrac{dy}{du} = 3u^{2}\) is the derivative of the outer function.
  • \(\dfrac{dy}{dx}\) is the derivative of the composite.

306.1.2 General

For \(T = \dfrac{1}{2}m\left(\dfrac{dx}{dt}\right)^{2}\) with \(x = x(t)\), the chain rule gives

\[ \dfrac{dT}{dt} = m\, \dfrac{dx}{dt}\, \dfrac{d^{2}x}{dt^{2}} \]

where

  • \(t\) is time.
  • \(x(t)\) is the displacement.
  • \(\dfrac{dx}{dt}\) is the velocity.
  • \(\dfrac{d^{2}x}{dt^{2}}\) is the acceleration.
  • \(m\) is the mass.
  • \(T\) is the kinetic energy.
  • \(\dfrac{dT}{dt}\) is the time derivative of \(T\).