17.7. EXERCISES 317

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 dv

dt = −3, and dzdt = 2. Find dP

dtwhen 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 respectto t.

1g

vdvdt

+dzdt

+1γ

dPdt

= 0.

Then when v = 7 and z = 8, finding dPdt involves nothing more than solving the following

for dPdt .

7g(−3)+2+

1γ

dPdt

= 0

ThusdPdt

= γ

(21g−2)

at this instant in time.

Example 17.7.2 In Bernoulli’s law above, each of v,z, and P are functions of (x,y,z), theposition of a point in the fluid. Find a formula for ∂P

∂x in terms of the partial derivatives ofthe other variables.

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

vg

vx + zx +1γ

Px = 0

and so

Px =−(

vvx + zxgg

)γ

Example 17.7.3 Suppose a level curve is of the form f (x,y) =C and that near a point onthis level curve y is a differentiable function of x. Find dy

dx .

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

fx + fydydx

= 0.

Solving for dydx gives

dydx

=− fx (x,y)

fy (x,y).

Example 17.7.4 Suppose a level surface is of the form f (x,y,z) =C. and that near a point(x,y,z) on this level surface z is a C1 function of x and y. Find a formula for zx.

This is an example of the use of the chain rule. Differentiate both sides of the equationwith respect to x. Since yx = 0,

fx + fzzx = 0.

Then solving for zx,

zx =− fx (x,y,z)

fz (x,y,z)