Connected rates of change
In a nutshell
One of the many uses of the chain rule is to relate multiple rates of change. Recall that a derivative gives a rate of change.
Example 1
The volume Vof a cylinder with a base circle with radius r (in centimetres) is given by the formula V=πr2h where h is the cylinder's height (also in centimetres). Given that the radius of this cylinder grows by 2 centimetres per second, find the rate of change of the volume when the radius is 5 centimetres. Assume that the height stays constant, at 10 cm.
Start by giving the rate of change of the radius with differentiation notation:
dtdr=2
where t is the time passed in seconds. Next find the rate of change of the volume with respect to the radius. This can be found by differentiating the equation given for the volume:
VdrdV=πr2h=2πrh
The question asks for the rate of change of the volume. Implicitly, this is with respect to time, thus you seek dtdV. The chain rule says that
dtdV=drdV×dtdr
Thus
dtdV=2πrh×2=4πrh
You have that h=10 and the question asks about when r=5:
dtdV=4π×5×10=200π≈628.32
Therefore, when the radius reaches 5 centimetres, the rate of change of the volume of the cylinder is about 628 cm3s−1(3s.f.). In other words, the volume increases by nearly 630 cm3 per second at the point when the radius is 5 cm.
Differential equations
Equations involving rates of change are called differential equations.
Example 2
The rate of change of the population P is directly proportional to the size of the population itself. Form a differential equation from this information.
You have that "the rate of change" refers to dtd, unless otherwise stated. Hence this information says
dtdP∝P
To turn this into an equation, you change the "∝" to "=" and multiply one side by a constant k:
dtdP=kP