Chain rules for partial differentiation

Multivariate functions: Partial derivatives

Chain rules for partial differentiation

The chain rule for differentiation describes what the derivative of a composition of two functions looks like. The composition of two functions #f# and #g# is the function #f\circ g# given by #f\circ g(t) = f(g(t))#.

Chain Rule of a function of a single variable

Before we discuss chain rules of multivariate functions, we briefly look back at the chain rule of functions of a single variable. Let #f# and #g# be two functions of one variable. The chain rule is #(f\circ g)' = (f'\circ g)\cdot g'#. In terms of functional rules, this reads \[\frac{\dd f(g(t))}{\dd t}=\frac{\dd f(x)}{\dd x}\left(g(t)\right)\cdot\frac{\dd g(t)}{\dd t}\tiny.\] The term # \frac{\dd f(x)}{\dd x}\left(g(t)\right)# represents the value of the function #\frac{\dd f(x)}{\dd x}# at #g(t)# and can also be written as #\left. \frac{\dd f(x)}{\dd x}\right|_{x=g(t)}#.

Often, a shorter notation is used. Write #x=g(t)#, and #y=f(x)#. Then #\frac{\dd y}{\dd x}# represents both the functional rule of #\frac{\dd f(x)}{\dd x}# and the value of #\frac{\dd f(x)}{\dd x}# in #x=g(t)#, and we can write the chain rule simply as \[\frac{\dd y}{\dd t}=\frac{\dd y}{\dd x}\cdot\frac{\dd x}{\dd t}\tiny.\] We will continue to work with this notation when we discuss chain rules for multivariate functions.

The function \[(3t^2-1)^5\] of #t# is composed of the functions #f# and #g# with functional rules \[f(x)=x^5\quad\text{and}\quad g(t)=3t^2-1\tiny.\] Indeed, #f\circ g(t)=f(g(t))=(g(t))^5=(3t^2-1)^5#.

The derivatives of #f# and #g# are #\frac{\dd f(x)}{\dd x}=5x^4# and #\frac{\dd g(t)}{\dd t}=6t# so, because of the chain rule, \[ \begin{array}{rcl}\frac{\dd(3t^2-1)^5}{\dd t}&=&\frac{\dd f(g(t))}{\dd t}\\&=&\left.\frac{\dd f(x)}{\dd x}\right|_{x=g(t)}\cdot\frac{\dd g(t)}{\dd t}\\ &=&\left.5x^4\right|_{x=3t^2-1}\cdot6t\\&=&5(3t^2-1)^4\cdot6t\\&=&30t\cdot(3t^2-1)^4\end {array}\]

In the short notation we write #y=f(x)=x^5# and #x=g(t)=3t^2-1#. Using the chain rule in the short form, we find:

\[ \begin{array}{rcl}\frac{\dd y}{\dd t}&=&\frac{\dd y}{\dd x}\cdot\frac{\dd x}{\dd t}\\ &=&\frac{\dd x^5}{\dd x}\cdot\frac{\dd (3t^2-1)}{\dd t}\\ &=&5x^4\cdot6t\\&=&5(3t^2-1)^4\cdot6t\\&=&30t\cdot(3t^2-1)^4\end {array}\]

Note that in #\frac{\dd y}{\dd t}# we use the variable #y# as a function of #t# and in #\frac{\dd y}{\dd x}# as a function of #x#.

We now discuss chain rules for functions of two variables.

Chain Rules for partial differentiation of bivariate functions

Let \(f(x,y)\) be a function of two variables #x# and #y#, so that #\frac{\partial f}{\partial x}# and #\frac{\partial f}{\partial y}# are continuous functions.

If \(x\) and \(y\) are differentiable functions of \(t\), then for \(f\) as a function of \(t\) holds: \[\frac{\dd f}{\dd t}=\frac{\partial f}{\partial x}\cdot\frac{\dd x}{\dd t}+ \frac{\partial f}{\partial y}\cdot\frac{\dd y}{\dd t}\]
If \(x\) and \(y\) are differentiable functions of two variables \(s\) and \(t\), then for \(f\) as a function of \(s\) and \(t\) holds: \[ \begin{array}{rcl}\frac{\partial f}{\partial s} &=&\frac{\partial f}{\partial x}\cdot\frac{\partial x}{\partial s}+ \frac{\partial f}{\partial y}\cdot\frac{\partial y}{\partial s} \\ \\ \frac{\partial f}{\partial t} &=&\frac{\partial f}{\partial x}\cdot\frac{\partial x}{\partial t}+ \frac{\partial f}{\partial y}\cdot\frac{\partial y}{\partial t}\end {array}\]

If the concept of continuity of functions like #\frac{\partial f}{\partial x}# and #\frac{\partial f}{\partial y}# is not known, then view it as a mild condition, which in our examples is almost always fulfilled.

When we talk about #f# as a function of #s# and #t#, we mean the composite function #f\circ \varphi#, where #\varphi(s,t)=\rv{x(s,t),y(s,t)}# is a mapping from a subset of #{\mathbb R}^2# to #{\mathbb R}^2#.

This double meaning of #f# (as a function of #x# and #y# and as a function of #s# and #t#) is also reflected in #x# and #y#, which are seen both as independent variables (e.g. in the denominators of expressions such as #\frac{\partial f}{\partial y}#), and as functions of #s# and #t# (e.g. in the numerator of #\frac{\partial y}{\partial t}#).

For functions of more than two variables similar chain rules exist:

General chain rule for partial differentiation of multivariate functions If \(w=w(x_1,x_2,\ldots,x_m)\) is a function of #m# variables #x_i# #(i=1,2,\ldots,m)# with continuous partial derivatives, and each \(x_i\) is a differentiable function of \(t_1,t_2,\ldots,t_n\), then, for each #j# #(1\le j\le n)#, the function \(w\) of \(t_1,t_2,\ldots,t_n\), has partial derivative \[\frac{\partial w}{\partial t_j}=\frac{\partial w}{\partial x_1}\cdot\frac{\partial x_1}{\partial t_j}+ \frac{\partial w}{\partial x_2}\cdot\frac{\partial x_2}{\partial t_j}+\cdots+\frac{\partial w}{\partial x_m}\cdot\frac{\partial x_m}{\partial t_j}\tiny.\]

Two more special cases:

If \(w=w(x)\) is a function of a single variable #x# with continuous partial derivative, and \(x\) is a differentiable function of \(s\) and \(t\), then \(w\), as a function of \(s\) and \(t\), has partial derivatives \[\frac{\partial w}{\partial s}=\frac{\dd w}{\dd x}\cdot\frac{\partial x}{\partial s}\quad\text{and}\quad\frac{\partial w}{\partial t}=\frac{\dd w}{\dd x}\cdot\frac{\partial x}{\partial t}\tiny.\]

If \(w=w(x,y,z)\) is a function of three variables with continuous partial derivatives, and \(x\), \(y\), and \(z\) are differentiable functions of \(t\), then \(w\), as a function of \(t\), has the derivative \[\frac{\dd w}{\dd t}=\frac{\partial w}{\partial x}\cdot\frac{\dd x}{\dd t}+ \frac{\partial w}{\partial y}\cdot\frac{\dd y}{\dd t}+\frac{\partial w}{\partial z}\cdot\frac{\dd z}{\dd t}\tiny.\]

Suppose \[z(x,y)=x\cdot y, \quad x=t^{7},\quad \text{and} \quad y=t^{9}\tiny.\] Then \(z\), viewed as a function of \(t\), is equal to \(t^{7}\cdot t^{9}=t^{16}\), so \(\frac{\dd z}{\dd t}=16 t^{15}\). This result also follows from the Chain Rule for partial differentiation. How?

\[\begin{array}{rcl} \frac{\dd z}{\dd t} &=&\frac{\partial z}{\partial x}\cdot\frac{\dd x}{\dd t}+ \frac{\partial z}{\partial y}\cdot\frac{\dd y}{\dd t} \\ &=&\frac{\partial \left(x\cdot y\right)}{\partial x}\cdot\frac{\partial\left( t^{7}\right)}{\partial t}+ \frac{\partial \left(x\cdot y\right)}{\partial y}\cdot\frac{\partial \left(t^{9}\right)}{\partial t} \\&=& y\cdot 7 t^{6}+x\cdot 9 t^{8} \\ &=& t^{9}\cdot 7 t^{6}+t^{7}\cdot 9 t^{8} \\&=& 16 t^{15}\end{array}\]

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