If is differentiable at the point and is differentiable at the point , then is differentiable at . Furthermore, let and , then
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(1) |
There are a number of related results that also go under the name of "chain rules." For example, if , , and , then
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(2) |
The "general" chain rule applies to two sets of functions
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(3) | |||
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(4) | |||
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(5) |
and
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(6) | |||
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(7) | |||
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(8) |
Defining the Jacobi rotation matrix by
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(9) |
and similarly for and , then gives
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(10) |
In differential form, this becomes
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(11) |
(Kaplan 1984).
See also
Derivative, Jacobian, Power Rule, Product Rule, Related Rates Problem Explore this topic in the MathWorld classroom
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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.
Cite this as:
Weisstein, Eric W. "Chain Rule." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ChainRule.html