You can compute a function’s slope by using the difference quotient. The difference quotient allows you to compute a slope if you don’t initially have two points to plug into the slope formula.
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 f (x) = x2 at the point (2, 4). Check out the first figure.
You can see the line drawn tangent to the curve at (2, 4). Because the slope of the tangent line is the same as the slope of the parabola at (2, 4), 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 (2, 4) — that you need for the slope formula.
Here’s how the inventors of calculus got around this roadblock. The next figure shows the tangent line again and a secant line intersecting the parabola at (2, 4) and at (10, 100).
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 (6, 36) and draw another secant using that point and (2, 4) again, as shown in the next figure.
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 (6, 36) and slid it down the parabola toward (2, 4), dragging the secant line along with it. Can you see that as the point gets closer and closer to (2, 4), 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 (x2,y2). As this point (x2,y2) slides closer and closer to (x1,y1), namely (2, 4), the run, which equals x1– x1, 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 (2, 4):
When the point (x2,y2) slides to (3, 9), the slope is
When the point slides to (2.1, 4.41), the slope is
When the point slides to (2.01, 4.0401), the slope is 4.01.
When the point slides to (2.001, 4.004001), the slope is 4.001.
Sure looks like the slope is headed toward 4.
As with all limit problems, the variable in this problem, x2approaches 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 (2, 4) — you’d get
which is undefined. But, of course, the slope at (2, 4) is precisely the slope you want — the slope of the line when the point does land on top of (2, 4). Herein lies the beauty of the limit process. With this limit, you get the exact slope of the tangent line at (2, 4) 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 f(x) at some number x= c, written as
is the slope of the tangent line to f drawn at c.
The slope fraction
is expressed with algebra terminology. Now you can 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,
when you write it in the fancy calculus way. A fraction is a quotient, right? And both y2 – y1 and x2 – x1 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.)
Okay, let’s lay out this process wherein
morphs into the difference quotient.
First, the run, x2 – x1 (in this example, x2 – 2), is called h. Next, because x1 = 2 and the run equals h,x2 equals 2 + h. You then write y1 as f(2) and y2 as f(2 + h). Making all the substitutions gives you the derivative of x2 at x = 2:
is simply the shrinking
stair step you can see in the previous figureas the point slides down the parabola toward (2, 4).
The next figure is basically the same as the previous one, except that instead of exact points like (6, 36) and (10, 100), the sliding point has the general coordinates of (2 + h, f (2 + h)), and the rise and the run are expressed in terms of h.
So this figure is the ultimate graph for
Are you confused by these two figures? Don’t sweat it. They both show the same thing. Both figures are visual representations of
Doing the math gives you, at last, the slope of the tangent line at (2, 4):
So the slope at the point (2, 4) is 4.
Main definition of the derivative: If you replace the point (2, f(2)) in the limit equation with the general point (x, f (x)), 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.
The next figure shows this general definition graphically. Note that this figure is virtually identical to the previous one, except that xs replace the 2s in the previous figure and that the moving point in this figure slides down toward any old point (x, f (x)) instead of toward the specific point (2, f(2)).
Now work out this limit and get the derivative for the parabola, f(x) = x2:
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.
The final figure sort of summarizes (in a simplified way) all the difficult preceding ideas about the difference quotient.
Summarizing the ideas
Like the previous three figures, the final figure contains a basic slope stair-step, a secant line, and a tangent line. The slope of the secant line is
The slope of the tangent line is
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
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