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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

(dy)/(dx)=(dy)/(du)·(du)/(dx).
(1)

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

(dz)/(dt)=(partialz)/(partialx)(dx)/(dt)+(partialz)/(partialy)(dy)/(dt).
(2)

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

((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)],
(9)

and similarly for (partialy_i/partialu_j) and (partialu_i/partialx_j), then gives

((partialy_i)/(partialx_j))=((partialy_i)/(partialu_i))((partialu_i)/(partialx_j)).
(10)

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 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.

Referenced on Wolfram|Alpha

Chain Rule

Cite this as:

Weisstein, Eric W. "Chain Rule." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ChainRule.html

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