This rate of change is constant. A straight line has one and only one slope.
straight line graph that relates them indicates constant speed. 45 miles per hour, say -- at every moment of time.
Let y = f(x). Let x be a specific point in the domain of f, and let z be another point. Then the coordinates of P are (x, f (x)), and the coordinates of Q are (z , f (z)). (Topic 4 of Precalculus.)
Therefore, the slope of the secant line PQ is
Let Δx approach 0 -- that is, let z approach x. Then the slope of the tangent line at P is
To be explicit, let us evaluate that limit for the function f(x) = x².
Then the coordinates of P are (x, x²), and the coordinates of Q are (z , z²).
First, we will evaluate the slope of the secant line PQ:
We first factored the difference of two squares (Lesson 19 of Algebra).
We then divided the numerator by z − x. (We may do that because z is never equal to x; that is, the denominator is never 0, even when we take the limit; Lesson 2).
We found the slope of the secant PQ to be z + x . To find the slope of the tangent at P, we will take the limit as z approaches x.
But z + x is a polynomial in z (Lesson 2). Therefore we may evaluate that limit simply by replacing z with x.
The slope, then, of the tangent to y = x² at the point x -- the rate of change of the function at that point -- is 2x. At x = 4, for example, the slope is 2· 4 = 8. At x = −5, the slope is 2· −5 = −10. And so on.
The value of the slope is itself a function of x. In this case it is 2x. 2x is the rate of change of f(x) = x² at the specific point x.
Now, since 2x was derived from f(x) = x², 2x is called the derived function, or the derivative, of x². To remind us that it came from f(x), we write f '(x) -- "f-prime of x."
The difference quotient then becomes
We will therefore express the definition of the derivative as follows.
Note: On taking the limit, the difference quotient is a function of h. x is constant. It is the specific point at which we calculate the rate of change of f(x).
In practice, we must simplify the difference quotient before taking the limit. We must express the numerator --
f (x + h) − f (x)
-- in such a way that we can divide it by h.
As an example, we will apply the definition to prove the following:
THEOREM. |
f(x) |
= |
x² |
|
implies |
|
|
|
f '(x) |
= |
2x. |
Proof. Here is the difference quotient, which we will proceed to simplfy:
1) |
|
(x + h)² − x² h |
|
2) |
= |
x² + 2xh + h² − x² h |
|
3) |
= |
2xh + h² h |
|
4) |
= |
2x + h. |
In going from line 1) to line 2), we squared the binomial x + h.
In going to line 3), we subtracted the x²s. That is, we subtracted f(x).
In going to line 4), we divided the numerator by h. (Lesson 20 of
Algebra.)
We can do that because h is never equal to 0, even when we take the limit (Lesson 2).
We now complete the defintion of the derivative and take the limit:
f '(x) |
= |
|
(2x + h) |
|
|
= |
2x. |
This is what we wanted to prove.
Whenever we apply the definition, we must express the difference quotient in such a way that we can evaluate the limit simply by replacing h with 0. For, that difference quotient -- in this case, 2x + h -- will be a continuous function of h.
Above are two examples. The function on the left does not have a derivative at x = 0, because the function is discontinuous there. At x = 0 there is obviously no tangent.
As for the function on the right, although it is continuous at x = 0, it is not differentiable. At an acute angular point, there is no tangent.
Thus, continuity is no guarantee of differentiability.
(Conversly, though, if a function is differentiable at a point -- if there is a tangent -- it will also be continuous there. The graph will be smooth and have no break.)
The graph on the right is the absolute value function. (Topic 5 of Precalculus.) Approaching 0 from the left, the slope of the tangent line -- which is the graph itself -- is −1. The slope approaching from the right, however, is +1. The limit at 0 -- which would be the derivative -- therefore does not exist . (Definition 2.2.)
Since differential calculus is the study of derivatives, it is fundamentally concerned with functions that are differentiable at all points of their domains. Such functions are called differentiable functions.
Can you name an elementary class of differentiable functions?
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Think about this yourself first!
Polynomials.
to take the derivative of what follows it. For example,
And so on.