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Chain Rule

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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).

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