Editing Chain rule (section)

===General rule: Vector-valued functions with multiple inputs===
The simplest way for writing the chain rule in the general case is to use the [[Total derivative#The total derivative as a linear map|total derivative]], which is a linear transformation that captures all [[directional derivative]]s in a single formula. Consider differentiable functions {{math|''f'' : '''R'''<sup>''m''</sup> → '''R'''<sup>''k''</sup>}} and {{math|''g'' : '''R'''<sup>''n''</sup> → '''R'''<sup>''m''</sup>}}, and a point {{math|'''a'''}} in {{math|'''R'''<sup>''n''</sup>}}. Let {{math|''D''<sub>'''a'''</sub> ''g''}} denote the total derivative of {{math|''g''}} at {{math|'''a'''}} and {{math|''D''<sub>''g''('''a''')</sub> ''f''}} denote the total derivative of {{math|''f''}} at {{math|''g''('''a''')}}. These two derivatives are linear transformations {{math|'''R'''<sup>''n''</sup> → '''R'''<sup>''m''</sup>}} and {{math|'''R'''<sup>''m''</sup> → '''R'''<sup>''k''</sup>}}, respectively, so they can be composed. The chain rule for total derivatives is that their composite is the total derivative of {{math|''f'' ∘ ''g''}} at {{math|'''a'''}}:
<math display="block">D_{\mathbf{a}}(f \circ g) = D_{g(\mathbf{a})}f \circ D_{\mathbf{a}}g,</math>
or for short,
<math display="block">D(f \circ g) = Df \circ Dg.</math>
The higher-dimensional chain rule can be proved using a technique similar to the second proof given above.<ref name="spivak_manifolds">{{cite book |first=Michael |last=Spivak |author-link=Michael Spivak |title=[[Calculus on Manifolds (book)|Calculus on Manifolds]] |location=Boston |publisher=Addison-Wesley |year=1965 |isbn=0-8053-9021-9 |pages=19–20 }}</ref>

Because the total derivative is a linear transformation, the functions appearing in the formula can be rewritten as matrices. The matrix corresponding to a total derivative is called a [[Jacobian matrix]], and the composite of two derivatives corresponds to the product of their Jacobian matrices. From this perspective the chain rule therefore says:
<math display="block">J_{f \circ g}(\mathbf{a}) = J_{f}(g(\mathbf{a})) J_{g}(\mathbf{a}),</math>
or for short,
<math display="block">J_{f \circ g} = (J_f \circ g)J_g.</math>

That is, the Jacobian of a composite function is the product of the Jacobians of the composed functions (evaluated at the appropriate points).

The higher-dimensional chain rule is a generalization of the one-dimensional chain rule. If {{mvar|k}}, {{mvar|m}}, and {{mvar|n}} are 1, so that {{math|''f'' : '''R''' → '''R'''}} and {{math|''g'' : '''R''' → '''R'''}}, then the Jacobian matrices of {{math|''f''}} and {{math|''g''}} are {{math|1 × 1}}. Specifically, they are:
<math display="block">\begin{align}
J_g(a) &= \begin{pmatrix} g'(a) \end{pmatrix}, \\
J_{f}(g(a)) &= \begin{pmatrix} f'(g(a)) \end{pmatrix}.
\end{align}</math>
The Jacobian of {{math|''f'' ∘ ''g''}} is the product of these {{math|1 × 1}} matrices, so it is {{math|''f''′(''g''(''a''))⋅''g''′(''a'')}}, as expected from the one-dimensional chain rule. In the language of linear transformations, {{math|''D''<sub>''a''</sub>(''g'')}} is the function which scales a vector by a factor of {{math|''g''′(''a'')}} and {{math|''D''<sub>''g''(''a'')</sub>(''f'')}} is the function which scales a vector by a factor of {{math|''f''′(''g''(''a''))}}. The chain rule says that the composite of these two linear transformations is the linear transformation {{math|''D''<sub>''a''</sub>(''f'' ∘ ''g'')}}, and therefore it is the function that scales a vector by {{math|''f''′(''g''(''a''))⋅''g''′(''a'')}}.

Another way of writing the chain rule is used when ''f'' and ''g'' are expressed in terms of their components as {{math|1='''y''' = ''f''('''u''') = (''f''<sub>1</sub>('''u'''), …, ''f''<sub>''k''</sub>('''u'''))}} and {{math|1='''u''' = ''g''('''x''') = (''g''<sub>1</sub>('''x'''), …, ''g''<sub>''m''</sub>('''x'''))}}. In this case, the above rule for Jacobian matrices is usually written as:
<math display="block">\frac{\partial(y_1, \ldots, y_k)}{\partial(x_1, \ldots, x_n)} = \frac{\partial(y_1, \ldots, y_k)}{\partial(u_1, \ldots, u_m)} \frac{\partial(u_1, \ldots, u_m)}{\partial(x_1, \ldots, x_n)}.</math>

The chain rule for total derivatives implies a chain rule for partial derivatives. Recall that when the total derivative exists, the partial derivative in the {{mvar|i}}-th coordinate direction is found by multiplying the Jacobian matrix by the {{mvar|i}}-th basis vector. By doing this to the formula above, we find:
<math display="block">\frac{\partial(y_1, \ldots, y_k)}{\partial x_i} = \frac{\partial(y_1, \ldots, y_k)}{\partial(u_1, \ldots, u_m)} \frac{\partial(u_1, \ldots, u_m)}{\partial x_i}.</math>
Since the entries of the Jacobian matrix are partial derivatives, we may simplify the above formula to get:
<math display="block">\frac{\partial(y_1, \ldots, y_k)}{\partial x_i} = \sum_{\ell = 1}^m \frac{\partial(y_1, \ldots, y_k)}{\partial u_\ell} \frac{\partial u_\ell}{\partial x_i}.</math>
More conceptually, this rule expresses the fact that a change in the {{math|''x''<sub>''i''</sub>}} direction may change all of {{math|''g''<sub>1</sub>}} through {{math|''g<sub>m</sub>''}}, and any of these changes may affect {{math|''f''}}.

In the special case where {{math|1=''k'' = 1}}, so that {{math|''f''}} is a real-valued function, then this formula simplifies even further:
<math display="block">\frac{\partial y}{\partial x_i} = \sum_{\ell = 1}^m \frac{\partial y}{\partial u_\ell} \frac{\partial u_\ell}{\partial x_i}.</math>
This can be rewritten as a [[dot product]]. Recalling that {{math|'''u''' {{=}} (''g''<sub>1</sub>, …, ''g''<sub>''m''</sub>)}}, the partial derivative {{math|∂'''u''' / ∂''x''<sub>''i''</sub>}} is also a vector, and the chain rule says that:
<math display="block">\frac{\partial y}{\partial x_i} = \nabla y \cdot \frac{\partial \mathbf{u}}{\partial x_i}.</math>

==== Example ====
Given {{math|1=''u''(''x'', ''y'') = ''x''<sup>2</sup> + 2''y''}} where {{math|1=''x''(''r'', ''t'') = ''r'' sin(''t'')}} and {{math|1=''y''(''r'',''t'') = sin<sup>2</sup>(''t'')}}, determine the value of {{math|∂''u'' / ∂''r''}} and {{math|∂''u'' / ∂''t''}} using the chain rule.{{citation needed|date=November 2023}}
<math display="block">\frac{\partial u}{\partial r}=\frac{\partial u}{\partial x} \frac{\partial x}{\partial r}+\frac{\partial u}{\partial y} \frac{\partial y}{\partial r} = (2x)(\sin(t)) + (2)(0) = 2r \sin^2(t),</math>
and
<math display="block">\begin{align}
\frac{\partial u}{\partial t}
&= \frac{\partial u}{\partial x} \frac{\partial x}{\partial t}+\frac{\partial u}{\partial y} \frac{\partial y}{\partial t} \\
&= (2x)(r\cos(t)) + (2)(2\sin(t)\cos(t)) \\
&= (2r\sin(t))(r\cos(t)) + 4\sin(t)\cos(t) \\
&= 2(r^2 + 2) \sin(t)\cos(t) \\
&= (r^2 + 2) \sin(2t).
\end{align}</math>

==== Higher derivatives of multivariable functions ====
{{Main|Faà di Bruno's formula#Multivariate version}}
Faà di Bruno's formula for higher-order derivatives of single-variable functions generalizes to the multivariable case. If {{math|1=''y'' = ''f''('''u''')}} is a function of {{math|1='''u''' = ''g''('''x''')}} as above, then the second derivative of {{math|''f'' ∘ ''g''}} is:
<math display="block">\frac{\partial^2 y}{\partial x_i \partial x_j} = \sum_k \left(\frac{\partial y}{\partial u_k}\frac{\partial^2 u_k}{\partial x_i \partial x_j}\right) + \sum_{k, \ell} \left(\frac{\partial^2 y}{\partial u_k \partial u_\ell}\frac{\partial u_k}{\partial x_i}\frac{\partial u_\ell}{\partial x_j}\right).</math>