Скачать или смотреть first principle I differentiation I ab-initio method I delta method I derivatives I class11 ,12

first principle I ab-initio method I delta method I differentiation I class 11, 12

The process of determining the derivative of a given function. This method is called differentiation from first principles or using the definition. We will now derive and understand the concept of the first principle of a derivative. This principle is the basis of the concept of derivative in calculus. A thorough understanding of this concept will help students apply derivatives to various functions with ease. We shall see that this concept is derived using algebraic methods. Nevertheless, the application of the concept remains irrelevant to its derivation methods. The main purpose of this discussion is to ensure that students have a clear idea of the concept of derivative. A first principle is a basic proposition or assumption that cannot be deduced from any other proposition or assumption. In philosophy, first principles are from First Cause attitudes and taught by Aristotelians, and nuanced versions of first principles are referred to as postulates by Kantians. In mathematics, first principles are referred to as axioms or postulates. In physics and other sciences, theoretical work is said to be from first principles, or ab initio, if it starts directly at the level of established science and does not make assumptions such as empirical model and parameter fitting. Differentiating a linear function A straight line has a constant gradient, or in other words, the rate of change of y with respect to x is a constant. graph of the straight line y = 3x + 2. We can calculate the gradient of this line as follows. We take two points and calculate the change in y divided by the change in x. When x changes from −1 to 0, y changes from −1 to 2, and so Differentiation from first principles of some simple curves For any curve it is clear that if we choose two points and join them, this produces a straight line. For different pairs of points we will get different lines, with very different gradients. Calculating the rate of change at a point We now explain how to calculate the rate of change at any point on a curve y = f(x). This is defined to be the gradient of the tangent drawn at that point The rate of change at a point P is defined to be the gradient of the tangent at P. NOTE: The gradient of a curve y = f(x) at a given point is defined to be the gradient of the tangent at that point. We use this definition to calculate the gradient at any particular point. Consider the graph below which shows a fixed point P on a curve. We also show a sequence of points Q1, Q2, . . . getting closer and closer to P. We see that the lines from P to each of the Q’s get nearer and nearer to becoming a tangent at P as the Q’s get nearer to P. The lines through P and Q approach the tangent at P when Q is very close to P. So if we calculate the gradient of one of these lines, and let the point Q approach the point P along the curve, then the gradient of the line should approach the gradient of the tangent at P, and hence the gradient of the curve. Example : We shall perform the calculation for the curve y = x2 at the point, P, where x = 3. The graph below shows the graph of y = x2 with the point P marked. We choose a nearby point Q and join P and Q with a straight line. We will choose Q so that it is quite close to P. Point R is vertically below Q, at the same height as point P, so that △PQR is right-angled. The graph of y = x2. P is the point (3, 9). Q is a nearby point. Suppose we choose point Q so that PR = 0.1. The x coordinate of Q is then 3.1 and its y coordinate is 3.12. Knowing these values we can calculate the change in y divided by the change in x and hence the gradient of the line PQ. The graph of y = x2. P is the point (x, y). Q is a nearby point. Point Q is chosen to be close to P on the curve. The x coordinate of Q is x + dx where dx is the symbol we use for a small change, or small increment in x. The corresponding change in y is written as dy. So the coordinates of Q are (x + dx, y + dy).





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