The Chain Rule

The chain rule is a formula for computing the derivative of the composition of two or more functions. That is, if $f$ is a function and $g$ is a function, then the chain rule expresses the derivative of the composite function $f \circ g (x) ≡ f [g (x)]$ in terms of the derivatives of $f$ and $g$. For example, the chain rule for $f \circ g$ is $\frac {df}{dx} = \frac {df}{dg} \, \frac {dg}{dx}$.

The chain rule above is for single variable functions $f(x)$ and $g(x)$. However, the chain rule can be generalized to functions with multiple variables. For example, consider a function $U$ with two variables $x$ and $y$: $U=U(x,y)$. $U$ could be electric potential energy at a location $(x,y)$. The motion of a test charge on the $xy$-plane can be described by $x=x(t)$, $y=y(t)$ where $t$ is a parameter representing time $t$. What we want to calculate is the rate of change of the potential energy $U$ as a function of time $t$. Assuming $x=x(t)$, $y=y(t)$, and $U=U(x,y)$ are all differentiable at $t$ and $(x,y)$, the chain rule is given as:

$\displaystyle{\frac{d U}{dt} = \frac{\partial U}{\partial x} \cdot \frac{dx}{dt} + \frac{\partial U}{\partial y} \cdot \frac{dy}{dt}}$

This relation can be easily generalized for functions with more than two variables.

Scalar Field

The chain rule can be used to take derivatives of multivariable functions with respect to a parameter.

Example

For $z = (x^2 + xy + y^2)^{1/2}$ where $x=x(t)$ and $y=y(t)$, express $\frac{dz}{dt}$ in terms of $\frac{dx}{dt}$and $\frac{dy}{dt}$:

$\displaystyle{\frac{dz}{dt} = \frac{d}{dt}(x^2 +xy+ y^2)^{1/2}}$

$\displaystyle{\,\,\,\quad= \frac{1}{2}(x^2 +xy + y^2)^{-1/2}\frac{d}{dt}(x^2 +xy+ y^2)}$

$\displaystyle{\,\,\,\quad=\frac{1}{2}(x^2 +xy+ y^2)^{-1/2}\left(\frac{d}{dt}(x^2) + \frac{d}{dt}(xy) +\frac{d}{dt}(y^2) \right)}$

$\displaystyle{\,\,\,\quad= \frac{ \left(x+\displaystyle{\frac{1}{2}} y \right)\displaystyle{\frac{dx}{dt}} + \left(y+\frac{1}{2} x \right) \displaystyle{\frac{dy}{dt}}}{\sqrt{x^2 +xy+ y^2}}}$