8—Multivariable Calculus 180In thermodynamics there are so many variables in use that there is a standard notation for a
partial derivative, indicating exactly which other variables are to be held constant.
(
∂U
∂V
)
Tand(
∂U
∂V
)
Prepresent the change in the internal energy of an object per change in volume during processes in which
respectively the temperature and the pressure are held constant. In the previous example with the
functionf=y, this says (
∂f
∂x
)
y= 0 and(
∂f
∂x
)
y′=− 1
This notation is a way to specify thedirectionin thex-yplane along which you’re taking the derivative.
8.2 Chain Rule
For functions of one variable, the chain rule allows you to differentiate with respect to still another
variable:ya function ofxandxa function oftallows
dy
dt
=
dy
dx
dx
dt
(8.3)
You can derive this simply from the definition of a derivative.
∆y
∆t
=
y
(
x(t+ ∆t)
)
−y
(
x(t)
)
∆t
=
y
(
x(t+ ∆t)
)
−y
(
x(t)
)
x(t+ ∆t)−x(t)
.x(t+ ∆t)−x(t)
∆t
=
∆y
∆x
.∆x
∆t
Take the limit of this product as∆t→ 0. Necessarily then you have that∆x→ 0 too (unless the
derivative doesn’t exist anyway). The second factor is then the definition of the derivativedx/dt, and
the first factor is the definition ofdy/dx. The Leibnitznotationas written in Eq. (8.3) leads you to
the required proof.
What happens with more variables? Roughly the same thing but with more manipulation, the
same sort of manipulation that you use to derive the rule for differentiating more complicated functions
of one variable (as in section1.5).
Computed
dt
f
(
x(t),y(t)
)
Back to the∆’s. The manipulation is much like the preceding except that you have to add and subtract
a term in the second line.