Multivariate Chain Rule
The differential of is . Along a path ,
Theorem 3.4 (Differential Form Of Chain Rule for Partial Derivatives)
Differential of a function is
Integral form
Theorem 3.5 (Integral Form Of Chain Rule for Partial Derivatives)
For a function , the change in between two endpoints is
Proof
Under ,
Taylor’s theorem gives
and
while
Hence
The limit is .
Related
Stated in
- Theorem 3.4 (Differential Form Of Chain Rule for Partial Derivatives)§3.3 Multivariate Chain Rule
- Theorem 3.5 (Integral Form Of Chain Rule for Partial Derivatives)§3.3 Multivariate Chain Rule