# concepts of differential calculus

This is graphically shown in Figure 5.7(b). Note that derivative of a function [Y=f (X)] is also written as d (fX) / dX. Meaning of the derivative in context: Applications of derivatives Straight … Points where f'(x) = 0 are called critical points or stationary points (and the value of f at x is called a critical value). %%EOF Content Guidelines 2. Before publishing your Articles on this site, please read the following pages: 1. 314 0 obj <> endobj Derivative of a Product of the Two Functions: Suppose Vis the product of the two separate functions f (X) and g(X). Privacy Policy3. %PDF-1.6 %���� Disclaimer Copyright, Share Your Knowledge Ramya is a consummate master of Mathematics, teaching college curricula. It is thus evident that derivative of a function shows the change in value of the dependent variable when change in the independent variable (∆X) becomes infinitesimally small. Let us take the following example of a power function which is of quadratic type: To illustrate it we have calculated the values of Y, associated with different values of X such as 1, 2, 2.5 and -1, -2, -2.5 and have been shown in Table 5.3. The derivative dY /dX or more precisely the first derivative of a function is defined as limit of the ratio ∆Y /∆X as ∆X approaches zero. At the limit of ∆Y/ ∆X when ∆X approaches zero, slope of the tangent such as tt at a point on a function becomes the derivative dY/dX of the function with respect to X. There are various types of functions and for them there are different rules for finding the derivatives. 9�U�\.�,��\$rzA�Jq��O=-�A�Q� C�Lg�͑�OL+��#�^�\��z�0Q�E�G��.��m&� Ʒ�ȡ��. You may need to revise this concept … It will be seen that a constant function is a horizontal straight line (having a zero slope) which shows that irrespective of the value of the variable X, the value of Y does not change at all. 7a The Concept of Limit of a Function (What must you know to learn Calculus?) To proceed with this booklet you will need to be familiar with the concept of the slope (also called the gradient) of a straight line. 0 Ramya is a consummate master of Mathematics, teaching college curricula. Thus. 149 7a.1 Introduction 149 7a.2 Useful Notations 149 7a.3 The Concept of Limit of a Function: Informal Discussion 151 7a.4 … Applications of derivatives. We have plotted the values of X and corresponding values of Y to get a U-shaped parabolic curve in Figure 5.8. As explained above, the derivatively of a function at a point measures the slope of the tangent at that point. Differential calculus is the calculus (which you can think of as a rule book for calculating things) of differentials. Our mission is to provide an online platform to help students to discuss anything and everything about Economics. Diﬀerential calculus is about describing in a precise fashion the ways in which related quantities change. This is graphed in Figure 5.7(a). Thus, if ∆X is infinitesimally small, ∆Y /∆X measures the slope of the function at a particular point and is called the derivative of the function with respect to X. Differential calculus deals with the rate of change of one quantity with respect to another. Ramya has been working as a private tutor for last 3 years. The constant ‘a’ implies that Y does not vary as X varies, that is. Suppose variable Vis a function of the variable U, that is, Y = f (U) and variable U is a function of variable X, that is, U = g (X). Similarly, if ∆X is reduced further, slope of the straight line between the two corresponding points will go on becoming closer and closer to the slope of the tangent tt drawn at point A to the curve. Consider Figure 5.6 when ∆X= X3 – X1, the slope of the corresponding straight line AB is equal to Y3-Y1/ X3-X1 becomes smaller and is equal to X2-X1, slope of that corresponding line AC is equal Y2-Y1/ X2-X1 . Share Your PDF File Welcome to EconomicsDiscussion.net! Thus, derivative dY/dX is slope of a function whether it is linear or non-linear and represents a change in the dependent variable due to a small change in the independent variable. Or you can consider it as a study of rates of change of quantities. Some other examples of power function and their derivatives are: It should be noted that any variable raised to the zero power (as in our example X0) is equal to 1. Derivative of the Quotient of the Two Functions: Suppose the variable 7is equal to the quotient of the two functions f (X) and g(X). Using the above rule for the derivative of a power function we have, dY / dX = 1 X 1.5 X1-1 = 1 X 1.5 X 0 = 1.5 X0 = 1.5. For example, velocity is the rate of change of distance with respect to time in a particular direction. Y is independent of X. h�bbd```b``��7@\$�f��" [@\$G�d�"Y�A\$��HX�9����I0,�� Vi\$�y,�&��H�p��@��^��3�!��`�t��?��G��=���p3�@� ��*� �� First, take the following power function: In this function 1.5 is the coefficient of variable X, that is, a and the power b of X is 1 (implicit). Derivative of Function of a Function (Chain Rule): When a variable Y is function of a variable U which in turn is related to another variable X, and if we wish to obtain a derivative of Y with respect to X, then we use chain rule for this purpose. As stated above, derivative of a function represents the change in the dependent variable due to a infinitesimally small change in the independent variable and is written as dY / dX for a function Y = f (X). Ramya has been working as a private tutor for last 3 years. If f is not assumed to be everywhere differentiable, then points at which it fails to be differentiable are also designated critical points. Where ‘a’ is constant. The derivative of this power function is equal to the power b multiplied by the coefficient a times the variable X raised to the power b – 1. BASIC CONCEPTS OF DIFFERENTIAL AND INTEGRAL CALCULUS 8.3 By definition x x 2x x ( x) x lim x (x x) x lim x f(x x) f(x) f(x) lim dx d 2 2 2 x 0 2 2 x 0 x 0 = lim (2x x) 2x 0 2x x 0 Thus, derivative of f(x) … Derivative of a Sum or Difference of Two Functions: The derivative of a sum of the two functions is equal to the sum of the derivatives obtained separately of the two functions. Then, to obtain the derivative of Y with respect to X, that is dY / dX, we first find the derivative of the two functions, Y = f(U) and U = g(X) separately and then multiply them together. Therefore, derivative dY/dX = 0. Process of finding the derivative of a function is called differentiation. Thus rule for the derivative of power function (Y = a Xb) is. It will be seen from Figure 5.6 that slope of line AC is more near to the slope of the tangent tt drawn at point A to the function curve. That is. It will be seen that derivative dY / dX or, in other words, slope of this quadratic function is changing at different values of X. Therefore, the derivative of a constant function is equal to zero. It will be seen from this figure that slope of the linear function (Y = 1.5 X) is constant and is equal to 1.5 over any range of the values of the variables X. Thus, Thus, according to the chain rule if Y= f(U) and U = g(X), then derivative of Y with respect X, can be obtained by multiplying together the derivative of Y with respect to U and the derivative of U with respect to X, Let us take some examples to illustrate this chain rule. The concept of a derivative is extensively used in economics and managerial decision making, especially in solving the problems of optimisation such as those of profit maximisation, cost minimisation, output and revenue maximisation.

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