Multivariate Calculus - Bruce E. Shapiro

82 LECTURE 12. THE CHAIN RULE
2. Draw an arrow from the u-node to each of its variable nodes, and then draw
an arrow from each of the variable nodes to the final node. Observe that by
following the arrows there are now two possible paths we can follow.
3. Label the arrows emanating from the original function with partial derivatives.
Label the arrows emanating from the variables themselves to the final variable
with ordinary derivatives.
4. Consider each possible pat from u to t, and from the product of derivatives as
you follow the path. The top path in figure 12.1, for example, gives
∂u dy
∂y dt
and the bottom path gives
∂u dx
∂x dt
5. Add together the products for each possible path. This gives
d ∂u dy
u(x, y) =
dt ∂y
dt + ∂u
∂x
dx
dt
Figure 12.2: Chain rule for a function of two variables
.
This method has an obvious generalization if there are more variables; for example,
if u = u(x, y, z), we add a third node labeled z and another path from u to
t through z. The result is
d
∂u dy
u(x, y, z) =
dt ∂y dt + ∂u
∂x
dx
dt + ∂u
∂z
Example 12.1 Let w = x 4 y 2 , x = sin t, y = t 2 . Find dw/dt using the chain rule.
Solution.
dw
dt
dz
dt
= ∂w dx
∂x dt + ∂w dy
∂y dt
= ∂
∂x (x4 y 2 ) d dt (sin t) + ∂ ∂y (x4 y 2 ) d dt (t2 )
= (4x 3 y 2 )(cos t) + x 4 (2y)(2t)
Revised December 6, 2006. Math 250, Fall 2006