Calculus For Dummies, 2nd Edition (2014)
Part IV. Differentiation
IN THIS PART …
The meaning of a derivative: It’s a slope and a rate — more specifically, a derivative tells you how fast y is changing compared to x.
How to calculate derivatives with the product rule, the quotient rule, and the chain rule.
Implicit differentiation, logarithmic differentiation, and the differentiation of inverse functions.
What a derivative tells you about the shape of a curve: Local minimums, local maximums, steepness, inflection points, concavity, critical numbers, and so on.
Differentiation word problems: Position, velocity, and acceleration, optimization, related rates, linear approximation, and tangent and normal lines.
Chapter 9. Differentiation Orientation
IN THIS CHAPTER
Discovering the simple algebra behind the calculus
Getting a grip on weird calculus symbols
Differentiating with Laurel and Hardy
Finding the derivatives of lines and curves
Tackling the tangent line problem and the difference quotient
Differential calculus is the mathematics of change and the mathematics of infinitesimals. You might say that it’s the mathematics of infinitesimal changes — changes that occur every gazillionth of a second.
Without differential calculus — if you’ve got only algebra, geometry, and trigonometry — you’re limited to the mathematics of things that either don’t change or that change or move at an unchanging rate. Remember those problems from algebra? One train leaves the station at 3 p.m. going west at 80 mph. Two hours later another train leaves going east at 50 mph … You can handle such a problem with algebra because the speeds or rates are unchanging. Our world, however, isn’t one of unchanging rates — rates are in constant flux.
Think about putting man on the moon. Apollo 11 took off from a moving launch pad (the earth is both rotating on its axis and revolving around the sun). As the Apollo flew higher and higher, the friction caused by the atmosphere and the effect of the earth’s gravity were changing not just every second, not just every millionth of a second, but every infinitesimal fraction of a second. The spacecraft’s weight was also constantly changing as it burned fuel. All of these things influenced the rocket’s changing speed. On top of all that, the rocket had to hit a moving target, the moon. All of these things were changing, and their rates of change were changing. Say the rocket was going 1,000 mph one second and 1,020 mph a second later — during that one second, the rocket’s speed literally passed through the infinite number of different speeds between 1,000 and 1,020 mph. How can you do the math for these ephemeral things that change every infinitesimal part of a second? You can’t do it without differential calculus.
And differential calculus is used for all sorts of terrestrial things as well. Much of modern economic theory, for example, relies on differentiation. In economics, everything is in constant flux. Prices go up and down, supply and demand fluctuate, and inflation is constantly changing. These things are constantly changing, and the ways they affect each other are constantly changing. You need calculus for this.
Differential calculus is one of the most practical and powerful inventions in the history of mathematics. So let’s get started already.
Differentiating: It’s Just Finding the Slope
Differentiation is the first of the two major ideas in calculus (the other is integration, which I cover in Part 5). Differentiation is the process of finding the derivative of a function like . The derivative is just a fancy calculus term for a simple idea you know from algebra: slope. Slope, as you know, is the fancy algebra term for steepness. And steepness is the fancy word for … No! Steepness is the ordinary word you’ve known since you were a kid, as in, “Hey, this road sure is steep.” Everything you study in differential calculus all relates back to the simple idea of steepness.
In differential calculus, you study differentiation, which is the process of deriving — that’s finding — derivatives. These are big words for a simple idea: Finding the steepness or slope of a line or curve. Throw some of these terms around to impress your friends. By the way, the root of the words differential and differentiation is difference — I explain the connection at the end of this chapter in the section on the difference quotient.
Consider Figure 9-1. A steepness of means that as the stickman walks one foot to the right, he goes up foot; where the steepness is 3, he goes up 3 feet as he walks 1 foot to the right. Where the steepness is zero, he’s at the top, going neither up nor down; and where the steepness is negative, he’s going down. A steepness of , for example, means he goes down 2 feet for every foot he goes to the right. This is shown more precisely in Figure 9-2.
FIGURE 9-1: Differentiating just means finding the steepness or slope.
FIGURE 9-2: The derivative = slope = steepness.
Negative slope: To remember that going down to the right (or up to the left) is a negative slope, picture an uppercase N, as shown in Figure 9-3.
FIGURE 9-3: This N line has a Negative slope.
Don’t be among the legions of students who mix up the slopes of vertical and horizontal lines. How steep is a flat, horizontal road? Not steep at all, of course. Zero steepness. So, a horizontal line has a slope of zero. (Like where the stick man is at the top of the hill in Figure 9-1.) What’s it like to drive up a vertical road? You can’t do it. And you can’t get the slope of a vertical line — it doesn’t exist, or, as mathematicians say, it’s undefined.
VARIETY IS THE SPICE OF LIFE
Everyone knows that . Now, wouldn’t it be weird if the next time you read this math fact, it was written as or ? How does grab you? Or ? Variety is not the spice of mathematics. When mathematicians decide on a way of expressing an idea, they stick to it — except, that is, with calculus. Are you ready? Hold on to your hat. All of the following are different symbols for the derivative — they all mean exactly the same thing: or or or or or or or or or or or . There are more. Now, you’ve got two alternatives: 1) Beat your head against the wall trying to figure out things like why some author uses one symbol one time and a different symbol another time, and what exactly does the d or f mean anyway, and so on and so on, or 2) Don’t try to figure it out; just treat these different symbols like words in different languages for the same idea — in other words, don’t sweat it. I strongly recommend the second option.
The slope of a line
Keep going with the slope idea — by now you should know that slope is what differentiation is all about. Take a look at the graph of the line, in Figure 9-4.
FIGURE 9-4: The graph of
You remember from algebra — I’m totally confident of this — that you can find points on this line by plugging numbers into x and calculating y: plug 1 into x and y equals 5, which gives you the point located at ; plug 4 into x and y equals 11, giving you the point , and so on.
I’m sure you also remember how to calculate the slope of this line. I realize that no calculation is necessary here — you go up 2 as you go over 1, so the slope is automatically 2. You can also simply note that is in slope-intercept form and that, since , the slope is 2. (See Chapter 5 if you want to review .) But bear with me because you need to know what follows. First, recall that
The rise is the distance you go up (the vertical part of a stair step), and the run is the distance you go across (the horizontal part of a stair step). Now, take any two points on the line, say, and , and figure the rise and the run. You rise up 10 from to because 5 plus 10 is 15 (or you could say that 15 minus 5 is 10). And you run across 5 from to because 1 plus 5 is 6 (or in other words, 6 minus 1 is 5). Next, you divide to get the slope:
Here’s how you do the same problem using the slope formula:
Plug in the points and :
Okay, let’s summarize what we know about this line. Table 9-1 shows six points on the line and the unchanging slope of 2.
TABLE 9-1 Points on the Line y = 2x + 3 and the Slope at Those Points
The derivative of a line
The preceding section showed you the algebra of slope. Now, here’s the calculus. The derivative (the slope) of the line in Figure 9-4 is always 2, so you write
Another common way of writing the same thing is
And you say,
· The derivative of the function, , is 2.
· (Read The derivative of the function, , is 2. That was a joke.)
The Derivative: It’s Just a Rate
Here’s another way to understand the idea of a derivative that’s even more fundamental than the concept of slope: A derivative is a rate. So why did I start the chapter with slope? Because slope is in some respects the easier of the two concepts, and slope is the idea you return to again and again in this book and any other calculus textbook as you look at the graphs of dozens and dozens of functions. But before you’ve got a slope, you’ve got a rate. A slope is, in a sense, a picture of a rate; the rate comes first, the picture of it comes second. Just like you can have a function before you see its graph, you can have a rate before you see it as a slope.
Calculus on the playground
Imagine Laurel and Hardy on a teeter-totter — check out Figure 9-5.
FIGURE 9-5: Laurel and Hardy — blithely unaware of the calculus implications.
Assuming Hardy weighs twice as much as Laurel, Hardy has to sit twice as close to the center as Laurel for them to balance. And for every inch that Hardy goes down, Laurel goes up two inches. So Laurel moves twice as much as Hardy. Voilà, you’ve got a derivative!
A derivative is a rate. A derivative is simply a measure of how much one thing changes compared to another — and that’s a rate.
Laurel moves twice as much as Hardy, so with calculus symbols you write
Loosely speaking, dL can be thought of as the change in Laurel’s position and dH as the change in Hardy’s position. You can see that if Hardy goes down 10 inches then dH is 10, and because dL equals 2 times dH, dL is 20 — so Laurel goes up 20 inches. Dividing both sides of this equation by dH gives you
And that’s the derivative of Laurel with respect to Hardy. (It’s read as, “dee L, dee H,” or as, “the derivative of L with respect to H.”) The fact that simply means that Laurel is moving 2 times as much as Hardy. Laurel’s rate of movement is 2 inches per inch of Hardy’s movement.
Now let’s look at it from Hardy’s point of view. Hardy moves half as much as Laurel, so you can also write
Dividing by dL gives you
This is the derivative of Hardy with respect to Laurel, and it means that Hardy moves inch for every inch that Laurel moves. Thus, Hardy’s rate is inch per inch of Laurel’s movement. By the way, you can also get this derivative by taking , which is the same as , and flipping it upside down to get .
These rates of 2 inches per inch and inch per inch may seem a bit odd because we often think of rates as referring to something per unit of time, like miles per hour. But a rate can be anything per anything. So, whenever you’ve got a this per that, you’ve got a rate; and if you’ve got a rate, you’ve got a derivative.
Speed — the most familiar rate
Speaking of miles per hour, say you’re driving at a constant speed of 60 miles per hour. That’s your car’s rate, and 60 miles per hour is the derivative of your car’s position, p, with respect to time, t. With calculus symbols, you write
This tells you that your car’s position changes 60 miles for each hour that the time changes. Or you can say that your car’s position (in miles) changes 60 times as much as the time changes (in hours). Again, a derivative just tells you how much one thing changes compared to another.
And just like the Laurel and Hardy example, this derivative, like all derivatives, can be flipped upside down:
This hours-per-mile rate is certainly much less familiar than the ordinary miles-per-hour rate, but it’s nevertheless a perfectly legitimate rate. It tells you that for each mile you go the time changes of an hour. And it tells you that the time (in hours) changes as much as the car’s position (in miles).
There’s no end to the different rates you might see. We just saw miles per hour and hours per mile. Then there’s miles per gallon (for gas mileage), gallons per minute (for water draining out of a pool), output per employee (for a factory’s productivity), and so on. Rates can be constant or changing. In either case, every rate is a derivative, and every derivative is a rate.
The rate-slope connection
Rates and slopes have a simple connection. All of the previous rate examples can be graphed on an x-y coordinate system, where each rate appears as a slope. Consider the Laurel and Hardy example again. Laurel moves twice as much as Hardy. This can be represented by the following equation:
Figure 9-6 shows the graph of this function.
FIGURE 9-6: The graph of
The inches on the H-axis indicate how far Hardy has moved up or down from the teeter-totter’s starting position; the inches on the L-axis show how far Laurel has moved up or down. The line goes up 2 inches for each inch it goes to the right, and its slope is thus , or 2. This is the visual depiction of , showing that Laurel’s position changes 2 times as much as Hardy’s.
One last comment. You know that . Well, you can think of dL as the rise and dH as the run. That ties everything together quite nicely.
Remember, a derivative is just a slope, and a derivative is also just a rate.
The Derivative of a Curve
The sections so far in this chapter have involved linear functions — straight lines with unchanging slopes. But if all functions and graphs were lines with unchanging slopes, there’d be no need for calculus. The derivative of the Laurel and Hardy function graphed previously is 2, but you don’t need calculus to determine the slope of a line. Calculus is the mathematics of change, so now is a good time to move on to parabolas, curves with changingslopes. Figure 9-7 is the graph of the parabola, .
FIGURE 9-7: The graph of
Notice how the parabola gets steeper and steeper as you go to the right. You can see from the graph that at the point , the slope is 1; at , the slope is 2; at , the slope is 3, and so on. Unlike the unchanging slope of a line, the slope of a parabola depends on where you are; it depends on the x-coordinate of wherever you are on the parabola. So, the derivative (or slope) of the function is itself a function of x — namely (I show you how I got that in a minute). To find the slope of the curve at any point, you just plug the x-coordinate of the point into the derivative, , and you’ve got the slope. For instance, if you want the slope at the point , plug 3 into the x, and the slope is times 3, or 1.5. Table 9-2 shows some points on the parabola and the steepness at those points.
TABLE 9-2 Points on the Parabola and the Slopes at Those Points
Here’s the calculus. You write
And you say,
The derivative of the function is .
Or you can say,
The derivative of is .
I promised to tell you how to derive this derivative of , so here you go:
1. Beginning with the original function, , take the power and put it in front of the coefficient.
2 times is , so that gives you .
3. Reduce the power by 1.
In this example, the 2 becomes a 1. So the derivative is or just .
This and many other differentiation techniques are discussed in Chapter 10.
The Difference Quotient
Sound the trumpets! You come now to what is perhaps the cornerstone of differential calculus: the difference quotient, the bridge between limits and the derivative. (But you’re going to have to be patient here, because it’s going to take me a few pages to explain the logic behind the difference quotient before I can show you what it is.) Okay, so here goes. I keep repeating — have you noticed? — the important fact that a derivative is just a slope. You learned how to find the slope of a line in algebra. In Figure 9-7, I gave you the slope of the parabola at several points, and then I showed you the short-cut method for finding the derivative — but I left out the important math in the middle. That math involves limits, and it takes us to the threshold of calculus. Hold on to your hat.
Slope is defined as , and .
To compute a slope, you need two points to plug into this formula. For a line, this is easy. You just pick any two points on the line and plug them in. But it’s not so simple if you want, say, the slope of the parabola at the point . Check out Figure 9-8.
FIGURE 9-8: The graph of (or) with a tangent line at
You can see the line drawn tangent to the curve at . Because the slope of the tangent line is the same as the slope of the parabola at , all you need is the slope of the tangent line to give you the slope of the parabola. But you don’t know the equation of the tangent line, so you can’t get the second point — in addition to — that you need for the slope formula.
Here’s how the inventors of calculus got around this roadblock. Figure 9-9 shows the tangent line again and a secant line intersecting the parabola at and at .
FIGURE 9-9: The graph of with a tangent line and a secant line.
Definition of secant line: A secant line is a line that intersects a curve at two points. This is a bit oversimplified, but it’ll do.
The slope of this secant line is given by the slope formula:
You can see that this secant line is steeper than the tangent line, and thus the slope of the secant, 12, is higher than the slope you’re looking for.
Now add one more point at and draw another secant using that point and again. See Figure 9-10.
FIGURE 9-10: The graph of with a tangent line and two secant lines.
Calculate the slope of this second secant:
You can see that this secant line is a better approximation of the tangent line than the first secant.
Now, imagine what would happen if you grabbed the point at and slid it down the parabola toward , dragging the secant line along with it. Can you see that as the point gets closer and closer to , the secant line gets closer and closer to the tangent line, and that the slope of this secant thus gets closer and closer to the slope of the tangent?
So, you can get the slope of the tangent if you take the limit of the slopes of this moving secant. Let’s give the moving point the coordinates . As this point slides closer and closer to , namely , the run, which equals , gets closer and closer to zero. So here’s the limit you need:
Watch what happens to this limit when you plug in four more points on the parabola that are closer and closer to :
· When the point slides to , the slope is , or 5.
· When the point slides to , the slope is , or 4.1.
· When the point slides to , the slope is 4.01.
· When the point slides to , the slope is 4.001.
Sure looks like the slope is headed toward 4. (By the way, the fact that the slope at — which you’ll see in a minute does turn out to be 4 — is the same as the y-coordinate of the point is a meaningless coincidence, as is the pattern you may have noticed in the above numbers between the y-coordinates and the slopes.)
As with all limit problems, the variable in this problem, approaches but never actually gets to the arrow-number (2 in this case). If it got to 2 — which would happen if you slid the point you grabbed along the parabola until it was actually on top of — you’d get , which is undefined. But, of course, the slope at is precisely the slope you want — the slope of the line when the point does land on top of . Herein lies the beauty of the limit process. With this limit, you get the exact slope of the tangent line at even though the limit function, , generates slopes of secant lines.
Here again is the equation for the slope of the tangent line:
And the slope of the tangent line is — you guessed it — the derivative.
Meaning of the derivative: The derivative of a function at some number , written as , is the slope of the tangent line to f drawn at c.
The slope fraction is expressed with algebra terminology. Now let’s rewrite it to give it that highfalutin calculus look. But first, finally, the definition you’ve been waiting for.
Definition of the difference quotient: There’s a fancy calculus term for the general slope fraction, or , when you write it in the fancy calculus way. A fraction is a quotient, right? And both and are differences, right? So, voilà, it’s called the difference quotient. Here it is:
(This is the most common way of writing the difference quotient. You may run across other, equivalent ways.) In the next two pages, I show you how morphs into the difference quotient.
Okay, let’s lay out this morphing process. First, the run, (in this example, ), is called — don’t ask me why — h. Next, because and the run equals h, equals . You then write as and as . Making all the substitutions gives you the derivative of at :
is simply the shrinking stair step you can see in Figure 9-10 as the point slides down the parabola toward .
Figure 9-11 is basically the same as Figure 9-10 except that instead of exact points like and , the sliding point has the general coordinates of , and the rise and the run are expressed in terms of h. Figure 9-11 is the ultimate figure for .
FIGURE 9-11: Graph of showing how a limit produces the slope of the tangent line at .
Have I confused you with these two figures? Don’t sweat it. They both show the same thing. Both figures are visual representations of . I just thought it’d be a good idea to show you a figure with exact coordinates before showing you Figure 9-11 with all that strange-looking f and h stuff in it.
Doing the math gives you, at last, the slope of the tangent line at :
So the slope at the point is 4.
Main definition of the derivative: If you replace the point in the limit equation above with the general point , you get the general definition of the derivative as a function of x:
So at last you see that the derivative is defined as the limit of the difference quotient.
Figure 9-12 shows this general definition graphically. Note that Figure 9-12 is virtually identical to Figure 9-11 except that xs replace the 2s in Figure 9-11 and that the moving point in Figure 9-12 slides down toward any old point instead of toward the specific point .
FIGURE 9-12: Graph of showing how a limit produces the slope of the tangent line at the general point
Now work out this limit and get the derivative for the parabola :
Thus for this parabola, the derivative (which is the slope of the tangent line at each value x) equals 2x. Plug any number into x, and you get the slope of the parabola at that x-value. Try it.
To close this section, let’s look at one final figure. Figure 9-13 sort of summarizes (in a simplified way) all the difficult preceding ideas about the difference quotient. Like Figures 9-10, 9-11, and 9-12, Figure 9-13 contains a basic slope stair-step, a secant line, and a tangent line. The slope of the secant line is , or . The slope of the tangent line is . You can think of as , and you can see why this is one of the symbols used for the derivative. As the secant line stair-step shrinks down to nothing, or, in other words, in the limit as and go to zero,
FIGURE 9-13: In the limit, .
Average Rate and Instantaneous Rate
Returning once again to the connection between slopes and rates, a slope is just the visual depiction of a rate: The slope, , just tells you the rate at which y changes compared to x. If, for example, the y-coordinate tells you distance traveled (in miles), and the x-coordinate tells you elapsed time (in hours), you get the familiar rate of miles per hour.
Each secant line in Figures 9-9 and 9-10 has a slope given by the formula . That slope is the average rate over the interval from . If y is in miles and x is in hours, you get the average speed in miles per hourduring the time interval from .
When you take the limit and get the slope of the tangent line, you get the instantaneous rate at the point . Again, if y is in miles and x is in hours, you get the instantaneous speed at the single point in time, . Because the slope of the tangent line is the derivative, this gives us another definition of the derivative.
Another definition of the derivative: The derivative of a function at some x-value is the instantaneous rate of change of f with respect to x at that value.
To Be or Not to Be? Three Cases Where the Derivative Does Not Exist
I want to discuss the three situations where a derivative fails to exist (see the “33333 Limit Mnemonic” section in Chapter 7). By now you certainly know that the derivative of a function at a given point is the slope of the tangent line at that point. So, if you can’t draw a tangent line, there’s no derivative — that happens in the first two cases below. In the third case, there’s a tangent line, but its slope and the derivative are undefined.
· There’s no tangent line and thus no derivative at any type of discontinuity: removable, infinite, or jump. (These types of discontinuity are discussed and illustrated in Chapter 7.) Continuity is, therefore, a necessary condition for differentiability. It’s not, however, a sufficient condition as the next two cases show. Dig that logician-speak.
· There’s no tangent line and thus no derivative at a sharp corner on a function (or at a cusp, a really pointy, sharp turn). See function f in Figure 9-14.
· Where a function has a vertical tangent line (which occurs at a vertical inflection point), the slope is undefined, and thus the derivative fails to exist. See function g in Figure 9-14. (Inflection points are explained in Chapter 11.)
FIGURE 9-14: Cases II and III where there’s no derivative.