Chain Rule

If $g(x)$ is Differentiable at the point $x$ and $f(x)$ is Differentiable at the point $g(x)$, then $f\circ g$ is Differentiable at $x$. Furthermore, let $y=f(g(x))$ and $u=g(x)$, then

$${dy\over dx}={dy\over du} \cdot {du\over dx}.$$ (1)

There are a number of related results which also go under the name of ``chain rules.'' For example, if $z=f(x,y)$, $x=g(t)$, and $y=h(t)$, then

$${dz\over dt}={\partial z\over\partial x}{dx\over dt}+{\partial z\over\partial y}{dy\over dt}.$$ (2)

The ``general'' chain rule applies to two sets of functions

$$y_1 = f_1(u_1, \ldots, u_p)$$

$$\vdots$$ (3)

$$y_m = f_m(u_1, \ldots, u_p)$$

and

$$u_1 = g_1(x_1,\ldots,x_n)$$

$$\vdots$$ (4)

$$u_p = g_p(x_1,\ldots,x_n).$$

Defining the $m\times n$ Jacobi Matrix by

$$\left({\partial y_i\over\partial x_j}\right)=\left[{\matrix{... ...m\over\partial x_n}\cr}}\right], \hrule width 0pt height 6.8pt$$ (5)

and similarly for $(\partial y_i/\partial u_j)$ and $(\partial u_i/\partial x_j)$ then gives

$$\left({\partial y_i\over\partial x_j}\right)=\left({\partial... ...rtial u_j}\right)\left({\partial u_i\over\partial x_j}\right).$$ (6)

In differential form, this becomes

$$dy_1=\left({{\partial y_1\over\partial u_1}{\partial u_1\ove... ...rtial u_p}{\partial u_p\over\partial x_2}}\right)\,dx_2+\ldots$$ (7)

(Kaplan 1984).

See also Derivative, Jacobian, Power Rule, Product Rule

References

Anton, H. Calculus with Analytic Geometry, 2nd ed. New York: Wiley, p. 165, 1984.

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.