12.7.1 Related Rates Problems
Sometimes several variables are related and, given information about how one variable is changing, you
want to find how the others are changing.
Example 12.7.1 Bernoulli’s law states that in an incompressible fluid,
where C is a constant. Here v is the speed, P is the pressure, and z is the height above some reference
point. The constants g and γ are the acceleration of gravity and the weight density of the fluid. Suppose
measurements indicate that
= 7 and z
= 8 in terms of g and
This is just an exercise in using the chain rule. Differentiate the two sides with respect to
Then when v = 7 and z = 8, finding
involves nothing more than solving the following for
at this instant in time.
Example 12.7.2 In Bernoulli’s law above, each of v,z, and P are functions of
, the position
of a point in the fluid. Find a formula for
in terms of the partial derivatives of the other variables.
This is an example of the chain rule. Differentiate both sides with respect to x.
Example 12.7.3 Suppose a level curve is of the form f
C and that near a point on this
level curve y is a differentiable function of x. Find
This is an example of the chain rule. Differentiate both sides with respect to x. This gives
Example 12.7.4 Suppose a level surface is of the form f
C. and that near a point
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 equation with respect to
x. Since yx = 0,
Then solving for zx,
Example 12.7.5 Polar coordinates are
Thus if f is a C1 scalar valued function you could ask to express fx in terms of the variables r and θ. Do
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
This will be done if you can find rx and θx. 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
Solving these using Cramer’s rule,
Hence fx in polar coordinates is