Editing Chain rule (section)

==== 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>