Chain Rule
المؤلف:
Anton
المصدر:
"The Chain Rule" and "Proof of the Chain Rule." §3.5 and AIII in Calculus with Analytic Geometry, 2nd ed. New York: Wiley
الجزء والصفحة:
...
14-5-2018
2318
Chain Rule
If 
is differentiable at the point 
and 
is differentiable at the point 
, then 
is differentiable at 
. Furthermore, let 
and 
, then
 |
(1)
|
There are a number of related results that also go under the name of "chain rules." For example, if 
, 
, and 
, then
 |
(2)
|
The "general" chain rule applies to two sets of functions
and
Defining the 
Jacobi rotation matrix by
![((partialy_i)/(partialx_j))=[(partialy_1)/(partialx_1) (partialy_1)/(partialx_2) ... (partialy_1)/(partialx_n); | | ... |; (partialy_m)/(partialx_1) (partialy_m)/(partialx_2) ... (partialy_m)/(partialx_n)],](http://mathworld.wolfram.com/images/equations/ChainRule/NumberedEquation3.gif) |
(9)
|
and similarly for 
and 
, then gives
 |
(10)
|
In differential form, this becomes
 |
(11)
|
(Kaplan 1984).
REFERENCES:
Anton, H. "The Chain Rule" and "Proof of the Chain Rule." §3.5 and AIII in Calculus with Analytic Geometry, 2nd ed. New York: Wiley, pp. 165-171 and A44-A46, 1999.
Apostol, T. M. "The Chain Rule for Differentiating Composite Functions" and "Applications of the Chain Rule. Related Rates and Implicit Differentiation." §4.10-4.11 in Calculus, 2nd ed., Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra. Waltham, MA: Blaisdell, pp. 174-179, 1967.
Kaplan, W. "Derivatives and Differentials of Composite Functions" and "The General Chain Rule." §2.8 and 2.9 in Advanced Calculus, 3rd ed. Reading, MA: Addison-Wesley, pp. 101-105 and 106-110, 1984.
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