calculus. We will assume we have a particular function of x, such that If x is assigned the value 10, the corresponding value of y will be (10)2 or 100. Now, if we increase the value of x by 2, making it 12, we may call this increase of 2 an increment or x. This results in an increase in the value of y, and we may call this increase an increment or y. From this we write"> Increments and Differentiation      Custom Search   INCREMENTS AND DIFFERENTIATION In this section we will extend our discussion of limits and examine the idea of the derivative, the basis of differential calculus. We will assume we have a particular function of x, such that If x is assigned the value 10, the corresponding value of y will be (10)2 or 100. Now, if we increase the value of x by 2, making it 12, we may call this increase of 2 an increment or x. This results in an increase in the value of y, and we may call this increase an increment or y. From this we write As x increases from 10 to 12, y increases from 100 to 144 so that and We are interested in the ratio because the limit of this ratio as x approaches zero is the derivative of y = f(X) As you recall from the discussion of limits, as x is made smaller, y gets smaller also. For our problem, the ratio approaches 20. This is shown in table 4-1. Table 4-l.-Slope Values We may use a much simpler way to find that the limit of as x approaches zero is, in this case, equal to 20. We have two equations and By expanding the first equation so that and subtracting the second from this, we have Dividing both sides of the equation by x gives Now, taking the limit as x approaches zero, gives Thus, NOTE: Equation (1) is one way of expressing the derivative of y with respect to x. Other ways are Equation (1) has the advantage that it is exact and true for all values of x. Thus if x=10 then and if x=3 then This method for obtaining the derivative of y with respect to x is general and may be formulated as follows: 1. Set up the function of x as a function of (x + x) and expand this function. 2. Subtract the original function of x from the new function of (x + x). 3. Divide both sides of the equation by x. 4.Take the limit of all the terms in the equation as x approaches zero. The resulting equation is the derivative of f(x) with respect to x.   Integrated Publishing, Inc. - A (SDVOSB) Service Disabled Veteran Owned Small Business