Chain Rule
If g(x) is differentiable at the point x and f(x) is differentiable at the point g(x), then f degreesg is differentiable at x. Furthermore, let y=f(g(x)) and u=g(x), then
There are a number of related results that also go under the name of "chain rules." For example, if z=f(x,y), x=g(t), and y=h(t), then
The "general" chain rule applies to two sets of functions
y_1 = f_1(u_1,...,u_p)
(3)
|
(4)
y_m = f_m(u_1,...,u_p)
(5)
and
u_1 = g_1(x_1,...,x_n)
(6)
|
(7)
u_p = g_p(x_1,...,x_n).
(8)
Defining the m×n Jacobi rotation matrix by
![画像: ((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)],](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/ChainRule/NumberedEquation3.svg){kind=link}
and similarly for (partialy_i/partialu_j) and (partialu_i/partialx_j), then gives
In differential form, this becomes
| dy_1=((partialy_1)/(partialu_1)(partialu_1)/(partialx_1)+...+(partialy_1)/(partialu_p)(partialu_p)/(partialx_1))dx_1+((partialy_1)/(partialu_1)(partialu_1)/(partialx_2)+...+(partialy_1)/(partialu_p)(partialu_p)/(partialx_2))dx_2+... |
(11)
|
(Kaplan 1984).
See also
Derivative, Jacobian, Power Rule, Product Rule, Related Rates Problem Explore this topic in the MathWorld classroomExplore with Wolfram|Alpha
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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.Referenced on Wolfram|Alpha
Chain RuleCite this as:
Weisstein, Eric W. "Chain Rule." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ChainRule.html