Some Applications of the Chain Rule
Read this section to learn how to apply the Chain Rule. Work through practice problems 1-8.
Parametric Equations
Suppose a robot has been programmed to move in the -plane so at time its coordinate will be and its coordinate will be . Both and are functions of the independent
parameter t, and , and the path of the robot (Fig. 4) can be found by plotting for lots of values of .
Typically we know and and need to find , the slope of the tangent line to the graph of . The Chain Rule says that , so, algebraically solving for , we get .
If we can calculate and , the derivatives of and with respect to the parameter , then we can determine , the rate of change of with respect to .
Example 8: Find the slope of the tangent line to the graph of when
Solution: and When , the object is at the point and the slope of the tangent line to the graph is .
Practice 5: Graph and find the slope of the tangent line when .
When we calculated , the slope of the tangent line to the graph of , we used the derivatives and , and each of these derivatives also has a geometric meaning:
measures the rate of change of with respect to - it tells us whether the -coordinate is increasing or decreasing as the t-variable increases.
measures the rate of change of with respect to .
Example 9: For the parametric graph in Fig. 5, tell whether and is positive or negative when .
Solution: As we move through the point (where ) in the direction of increasing values of , we are moving to the left so is decreasing and is negative.
Similarly, the values of are increasing so is positive. Finally, the slope of the tangent line, , is negative.
(As check on the sign of we can also use the result = .)
Practice 6: For the parametric graph in the previous example, tell whether and is positive or negative when and when .