Sometimes several variables are related and, given information about how one variable is
changing, you want to find how the others are changing.

Example 21.7.1Bernoulli’s law states that in an incompressible fluid,

2
-v + z + P = C
2g γ

where C is a constant. Here v is the speed, P is the pressure, and z is the height abovesome reference point. The constants g and γ are the acceleration of gravity and the weightdensity of the fluid. Suppose measurements indicate that

ddvt

= −3, and

ddzt

= 2. Find

ddPt

when v = 7 and z = 8 in terms of g and γ.

This is just an exercise in using the chain rule. Differentiate the two sides with respect
to t.

1v dv+ dz + 1-dP-= 0.
g dt dt γ dt

Then when v = 7 and z = 8, finding

ddPt

involves nothing more than solving the following
for

dPdt

.

7 1-dP-
g (− 3) + 2+ γ dt = 0

Thus

dP ( 21 )
---= γ --− 2
dt g

at this instant in time.

Example 21.7.2In Bernoulli’s law above, each of v,z, and P are functions of

(x,y,z)

, the position of a point in the fluid. Find a formula for

∂∂Px-

in terms of thepartial derivatives of the other variables.

This is an example of the chain rule. Differentiate both sides with respect to
x.

v 1-
gvx + zx + γPx = 0

and so

( vv + z g)
Px = − --x---x-- γ
g

Example 21.7.3Suppose a level curve is of the form f

(x,y)

= C and that neara point on this level curve y is a differentiable function of x. Find

ddyx

.

This is an example of the chain rule. Differentiate both sides with respect to x. This
gives

fx + fydy-= 0.
dx

Solving for

dy
dx

gives

dy − f (x,y)
---= ---x-----.
dx fy(x,y)

Example 21.7.4Suppose a level surface is of the form f

(x,y,z)

= C. and thatnear a point

(x,y,z)

on this level surface z is a C^{1}function of x and y. Find aformula for z_{x}.

This is an example of the use of the chain rule. Differentiate both sides of the
equation with respect to x. Since y_{x} = 0,

fx +fzzx = 0.

Then solving for z_{x},

z = −-fx(x,y,z)
x fz(x,y,z)

Example 21.7.5Polar coordinates are

x = rcosθ,y = rsinθ. (21.13)

(21.13)

Thus if f is a C^{1}scalar valued function you could ask to express f_{x}in terms of thevariables r and θ. Do so.

This is an example of the chain rule. Abusing notation slightly, regard f as a function
of position in the plane. This position can be described with any set of coordinates. Thus
f

(x,y)

= f

(r,θ)

and so

f = f r + f θ .
x r x θ x

This will be done if you can find r_{x} and θ_{x}. However you must find these in terms of r
and θ, not in terms of x and y. Using the chain rule on the two equations for the
transformation in 21.13,