Steps for Related Rates Problems:
1. Figure out what rate you want and what rate you have been given. For example, suppose the
question gave you
dx
dy
and asked you to calculate
.
dt
dt
2. Now you need to determine the relationship between the two variables, say, x and y. To do that,
reread the problem and/or draw a diagram.
3. Now you have two options: either use implicit differentiation or the chain rule.
•
Implicit: differentiate your equation from step (2) with respect to t to find the relationship
between the rates
•
dy
dx
and
.
dt
dt
Chain rule: set up an equation which uses the chain rule to calculate the rate you want from
the rate you have. For the rates given above, you’d write
your equation for y with respect to x to find
dy dy dx
. Now differentiate
=
dt dx dt
dy
.
dx
4. Substitute the values into the relationship you found in part 3 and calculate. Done!
Example: the ladder problem.
A ladder is slipping down a vertical wall. The ladder is 5 m long and the top of it is slipping down the
wall at a constant rate of 0.3 m/s. How fast is the bottom of the ladder moving away from the wall when
the top of the ladder is 4 m above the ground?
(Handwritten solution is available separately.)