Combining Differentiation Rules
As we have seen throughout the examples in this section, it seldom happens that we are called on to apply just one differentiation rule to find the derivative of a given function. At this point, by combining the differentiation rules, we may find the derivatives of any polynomial or rational function. Later on we will encounter more complex combinations of differentiation rules. A good rule of thumb to use when applying several rules is to apply the rules in reverse of the order in which we would evaluate the function.
Example 3.28
Combining Differentiation Rules
For \(k \left(\right. x \left.\right) = 3 h \left(\right. x \left.\right) + x^{2} g \left(\right. x \left.\right) ,\) find \(k^{'} \left(\right. x \left.\right) .\)
Solution
Finding this derivative requires the sum rule, the constant multiple rule, and the product rule.
Example 3.29
Extending the Product Rule
For \(k \left(\right. x \left.\right) = f \left(\right. x \left.\right) g \left(\right. x \left.\right) h \left(\right. x \left.\right) ,\) express \(k^{'} \left(\right. x \left.\right)\) in terms of \(f \left(\right. x \left.\right) , g \left(\right. x \left.\right) , h \left(\right. x \left.\right) ,\) and their derivatives.
Solution
We can think of the function \(k \left(\right. x \left.\right)\) as the product of the function \(f \left(\right. x \left.\right) g \left(\right. x \left.\right)\) and the function \(h \left(\right. x \left.\right) .\) That is, \(k \left(\right. x \left.\right) = \left(\right. f \left(\right. x \left.\right) g \left(\right. x \left.\right) \left.\right) \cdot h \left(\right. x \left.\right) .\) Thus,
Example 3.30
Combining the Quotient Rule and the Product Rule
For \(h \left(\right. x \left.\right) = \frac{2 x^{3} k \left(\right. x \left.\right)}{3 x + 2} ,\) find \(h^{'} \left(\right. x \left.\right) .\)
Solution
This procedure is typical for finding the derivative of a rational function.
Checkpoint 3.19
Find \(\frac{d}{d x} \left(\right. 3 f \left(\right. x \left.\right) - 2 g \left(\right. x \left.\right) \left.\right) .\)
Example 3.31
Determining Where a Function Has a Horizontal Tangent
Determine the values of \(x\) for which \(f \left(\right. x \left.\right) = x^{3} - 7 x^{2} + 8 x + 1\) has a horizontal tangent line.
Solution
To find the values of \(x\) for which \(f \left(\right. x \left.\right)\) has a horizontal tangent line, we must solve \(f^{'} \left(\right. x \left.\right) = 0 .\) Since
we must solve \(\left(\right. 3 x - 2 \left.\right) \left(\right. x - 4 \left.\right) = 0 .\) Thus we see that the function has horizontal tangent lines at \(x = \frac{2}{3}\) and \(x = 4\) as shown in the following graph.
Example 3.32
Finding a Velocity
The position of an object on a coordinate axis at time \(t\) is given by \(s \left(\right. t \left.\right) = \frac{t}{t^{2} + 1} .\) What is the initial velocity of the object?
Solution
Since the initial velocity is \(v \left(\right. 0 \left.\right) = s^{'} \left(\right. 0 \left.\right) ,\) begin by finding \(s^{'} \left(\right. t \left.\right)\) by applying the quotient rule:
After evaluating, we see that \(v \left(\right. 0 \left.\right) = 1 .\)
Checkpoint 3.20
Find the values of \(x\) for which the graph of \(f \left(\right. x \left.\right) = 4 x^{2} - 3 x + 2\) has a tangent line parallel to the line \(y = 2 x + 3 .\)
Student Project
Formula One Grandstands
Formula One car races can be very exciting to watch and attract a lot of spectators. Formula One track designers have to ensure sufficient grandstand space is available around the track to accommodate these viewers. However, car racing can be dangerous, and safety considerations are paramount. The grandstands must be placed where spectators will not be in danger should a driver lose control of a car (Figure 3.20).
Safety is especially a concern on turns. If a driver does not slow down enough before entering the turn, the car may slide off the racetrack. Normally, this just results in a wider turn, which slows the driver down. But if the driver loses control completely, the car may fly off the track entirely, on a path tangent to the curve of the racetrack.
Suppose you are designing a new Formula One track. One section of the track can be modeled by the function \(f \left(\right. x \left.\right) = x^{3} + 3 x^{2} + x\) (Figure 3.21). The current plan calls for grandstands to be built along the first straightaway and around a portion of the first curve. The plans call for the front corner of the grandstand to be located at the point \(\left(\right. −1.9 , 2.8 \left.\right) .\) We want to determine whether this location puts the spectators in danger if a driver loses control of the car.
- Physicists have determined that drivers are most likely to lose control of their cars as they are coming into a turn, at the point where the slope of the tangent line is 1. Find the \(\left(\right. x , y \left.\right)\) coordinates of this point near the turn.
- Find the equation of the tangent line to the curve at this point.
- To determine whether the spectators are in danger in this scenario, find the x-coordinate of the point where the tangent line crosses the line \(y = 2.8 .\) Is this point safely to the right of the grandstand? Or are the spectators in danger?
- What if a driver loses control earlier than the physicists project? Suppose a driver loses control at the point \(\left(\right. −2.5 , 0.625 \left.\right) .\) What is the slope of the tangent line at this point?
- If a driver loses control as described in part 4, are the spectators safe?
- Should you proceed with the current design for the grandstand, or should the grandstands be moved?
This lesson is part of:
Derivatives