If we differentiate a position function at a given time, we obtain the velocity at that time. It seems reasonable to conclude that knowing the derivative of the function at every point would produce valuable information about the behavior of the function. However, the process of finding the derivative at even a handful of values using the techniques of the preceding section would quickly become quite tedious.
In this section we define the derivative function and learn a process for finding it. The derivative function gives the derivative of a function at each point in the domain of the original function for which the derivative is defined.
We can formally define a derivative function as follows. Let be a function. The derivative function , denoted by , is the function whose domain consists of those values of such that the following limit exists:. A function is said to be differentiable at if exists. More generally, a function is said to be differentiable on if it is differentiable at every point in an open set , and a differentiable function is one in which exists on its domain.
In the next few examples we use Figure to find the derivative of a function. Find the derivative of. Start directly with the definition of the derivative function. Use Figure. Find the derivative of the function. Follow the same procedure here, but without having to multiply by the conjugate.
Use Figure and follow the example. We use a variety of different notations to express the derivative of a function. In Figure we showed that if , then. If we had expressed this function in the form , we could have expressed the derivative as or.
We could have conveyed the same information by writing. Thus, for the function , each of the following notations represents the derivative of :. In place of we may also use Use of the notation called Leibniz notation is quite common in engineering and physics.
To understand this notation better, recall that the derivative of a function at a point is the limit of the slopes of secant lines as the secant lines approach the tangent line. The slopes of these secant lines are often expressed in the form where is the difference in the values corresponding to the difference in the values, which are expressed as Figure. Thus the derivative, which can be thought of as the instantaneous rate of change of with respect to , is expressed as.
We have already discussed how to graph a function, so given the equation of a function or the equation of a derivative function, we could graph it. Given both, we would expect to see a correspondence between the graphs of these two functions, since gives the rate of change of a function or slope of the tangent line to.
In Figure we found that for. If we graph these functions on the same axes, as in Figure , we can use the graphs to understand the relationship between these two functions. First, we notice that is increasing over its entire domain, which means that the slopes of its tangent lines at all points are positive. Consequently, we expect for all values of in its domain. Furthermore, as increases, the slopes of the tangent lines to are decreasing and we expect to see a corresponding decrease in.
We also observe that is undefined and that , corresponding to a vertical tangent to at 0. The graphs of these functions are shown in Figure. Observe that is decreasing for. For these same values of. For values of is increasing and. Also, has a horizontal tangent at and. Use the following graph of to sketch a graph of. The solution is shown in the following graph.
Observe that is increasing and on. Also, is decreasing and on and on. Also note that has horizontal tangents at -2 and 3, and and. Sketch the graph of. On what interval is the graph of above the -axis? The graph of is positive where is increasing.
First, we consider the relationship between differentiability and continuity. We will see that if a function is differentiable at a point, it must be continuous there; however, a function that is continuous at a point need not be differentiable at that point. In fact, a function may be continuous at a point and fail to be differentiable at the point for one of several reasons.
Let be a function and be in its domain. If is differentiable at , then is continuous at. If is differentiable at , then exists and. We want to show that is continuous at by showing that. Therefore, since is defined and , we conclude that is continuous at. We have just proven that differentiability implies continuity, but now we consider whether continuity implies differentiability.
To determine an answer to this question, we examine the function. This function is continuous everywhere; however, is undefined. This observation leads us to believe that continuity does not imply differentiability. For ,. See Figure. Consider the function :.
Thus does not exist. A quick look at the graph of clarifies the situation. The function has a vertical tangent line at 0 Figure. The function also has a derivative that exhibits interesting behavior at 0.
By Jane. For example: If y is a function of x, then we sometimes write the derivative of y with respect to x as the following: When we write indefinite integrals, they are written as:. Now, we can see where the notation for the integral comes from. The bounds of integration from a to b are like the first and last x values for the summation.
Health Professions. Law School. Graduate School. Middle School. Related Content November 1, Working out your brain. That is the question. October 13, Dreaming and designing: a short guide to your many lives. October 11, The role of insurance and common threats in health insurance markets. September 27, How to solve an empirical formula problem. The metaphysics of the differential calculus is the most important to treat here. This metaphysics, of which so much has been written, is even more important, and perhaps as difficult to develop as these same rules of the calculus.
Several geometers, among them, Monsieur Rolle, unable to admit the supposition that there are infinitely small magnitudes, rejected it completely, and claimed that this principle was faulty and capable of inducing error [6]. Monsieur Leibniz complicated the objections that he felt might be made against infinitely small quantities, as considered by the differential calculus, having preferred to reduce his infinitely small to the not-comparable, which ruined the geometric exactitude of the calculus.
And what weight, according to Monsieur Fontenelle, must not the authority of the inventor have against the invention? Others, namely Monsieur Nieuwentyt [8] , admit differentials of the first order alone, and reject all those of higher order — without foundation. Since imagining in a circle an infinitely small cord of the first order, an abscissa or sinus, and the corresponding line of the second, infinitely small; if the cord of the second is infinitely small, the abscissa of the fourth is infinitely small, etc.
This is easily demonstrated by elementary Geometry, since the diameter of a finite circle is always in relation to the chord as the chord is to the corresponding abscissa. So if we admit, one time, the infinitely small of the first order, all others become necessary. Our remarks here are only to show that, in admitting the infinitely small of the first order, we must admit all others to infinity, because one can after all, easily pass from the metaphysics of infinity into the differential calculus, as we demonstrate further below [9].
Monsieur Newton subscribes to another principle, and we can say that the metaphysics of this great geometer on the calculus of fluxions is very exact and very enlightening lumineuse , though he'd be satisfied to give us just a glimpse. He has never looked at the differential calculus as the calculus of infinitely small quantities, but as the method of first and second reasons [10] , that is to say, the method of finding the limits of relations.
So, this illustrious author has never differentiated using quantities, only equations, because all equations contain a relation between two variables, and the differentiation of equations only consists in finding the limits of relation between finite differences of two variables that the equation contains [11].
This is what we must clarify with an example that will simultaneously give us the clearest idea and the most exact demonstration of the method of the differential calculus.
Let A M figure 3. Let us propose to draw a tangent M Q from this parabola at point M. Suppose that the problem is solved, and imagine an ordinate p m at any finite distance of P M; and for the points M, m, we draw the line m M R. It is evident that: 1.
Therefore, the relation is the limit of the relation of m O to O M [15]. Thus, if the limit of the relation between m O to O M is found, expressed algebraically, we obtain the algebraic expression of the relation of MP to PQ and, consequently, the algebraic expression of the ordinate relation at the subtangent, where this sub tangent will be found [16].
So is in general the relation of m O to O M, at any part for which we take point m. This relation is always smaller than , but the smaller z , the more this relation will be augmented.
And, as we can extend z as little as is desired, we can approach the relation as close as we want to the relation. Thus, is the limit of the relation of , which is to say of the relation.
So, is equal to , which we have found also to be the limit of the relation of m O to OM , because two magnitudes which make the limit of the same magnitude, are necessarily equal between them [19]. By hypothesis, the quantity Y can approach Z as closely as we desire.
That is to say that the difference between Y and X can be as small as wished. Therefore, since Z differs from X by the quantity V , it follows that Y cannot approach Z any closer than the quantity V, and consequently, that of Z is not the limit of Y , which is contrary to the hypothesis [20].
Limit , Exhaustion. Thus, the result is that is equal to. What does that mean? There is no absurdity in this, because can be equal to anything desired: it therefore can be. This limit is the quantity of which the relation keeps approaching in supposing z and u both real and decreasing, and of which this relation approaches almost anything desired.
See Limit , Series , Progression , etc. One gets the impression from all this, that we intend to say that the method of the differential calculus gives us exactly the same relation that comes from that given by the preceding calculus.
Other examples become even more complicated. The latter appears to us to suffice to make the true metaphysics of the differential calculus initially understood. When understood well once, the supposition that one has made of infinitely small quantities will be felt to be only for abridging and simplifying reasoning.
However, as the foundation of the differential calculus does not necessarily presume the existence of these quantities, as the calculus only consists in algebraically determining the limit of a relation that has already been expressed in lines, and in equaling these two limits, which allows us to find one of the lines for which we search [27]. This definition is, perhaps, the most precise and clearest that can be given for the differential calculus.
Still, it cannot be very well understood when the calculus has been made familiar, because often the true definition of a science can only be very sensible to those who have studied science. See Preliminary Discourse, p.
In the preceding example, the known geometric limit of the relation of z to u is the relation of the ordinate to the subtangent. With the differential calculus is sought the algebraic limit of the relation of z to u , and is found.
Thus naming s the subtangent, we have. It thus will suffice to make more familiar in the example above, the tangents of the parabola, and as the entire differential calculus can be reduced to a problem of tangents, it follows that the preceding principles may always be applied to different problems that are resolved by the calculus, as the invention of maxima and minima , points of inflection and folding rebroussement , Etc.
See these words. How, in effect, can one find the maximum or minimum? It is, evidently, to give the difference of dy equal to zero or to infinity, but to speak more exactly, it is to search for the quantity , which expresses the limit of the relation of a finite dy to a finite dx , and then makes this quantity either nothing or infinite.
And there, the mystery is completely explained. It's : that's to say that we look for the value of x which makes the limit of the relation of finite dy to finite dx infinite. We've seen above that there is no clean point of infinitely small quantities of the first order within the differential calculus, that the quantities that one therefore names are supposed to be divided by other quantities which are supposed to be infinitely small, and that in this state, they mark, not infinitely small quantities, nor even fractions, of which the numerator and the denominator are infinitely small, but the limit of a relation between two finite quantities.
There are even second differences, and others of a more elevated order. In geometry, there is no true point d d y , but when d d y is encountered in an equation, it is supposed to be divided by a quantity, dx 2 , or another of the same order. In this state, what is? It is the limit of the relation, , divided by d x , or what will be still clearer, it is, in making a finite quantity, the limit.
The differentio-differential calculus is the method of differentiating differential magnitudes, and the differentio-differential quantity is called the differential of a differential.
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