The rate of change of y with respect to x for a given value of x is therefore We are thus led to find the limit of 2ax + a Ax, a function of Ax, as Ar approaches zero. In computing this limit, x has a given value and is regarded as constant. In order to prove the assumption made earlier that the limit is 2ax, it must be shown that a Ax, the difference between the variable 2ax + a Ax and the constant 2ax, can be made as small as we please by making Ax sufficiently small. This follows readily. For if we wish to make a Ax as small as 0.001, it is sufficient to take Ax = 0.001/a, which is possible since Ax approaches zero and can therefore be made as small as we please. The limits encountered in computing lim Ay where У is a polynomial or a rational function of x, can be computed by means of the following theorems, which we assume without proof. Theorem 1. The limit of the sum of several variables is the sum of their limits. Theorem 2. The limit of the product of several variables is the product of their limits. Theorem 3. The limit of the quotient of two variables is the quotient of their limits, provided the limit of the divisor is not zero. The limit of the difference of two variables may be found by Theorem 1, since u v = u + (− v). = In applying these theorems, it is frequently convenient to regard y c, a constant, as a function of x, whose limit, as x approaches a, is c. Give the details of the computation of the limit of Ay/ Ax as Ax approaches zero. = lim (3x2 + 3x Ax + Ax2) Ax÷0 lim 3x2 + lim 3x Ax + lim Ax Ax (Theorem 1) 93. Derivative of a Function. DEFINITION. If y is a function of x, and if Ay is the change in y corresponding to a change of Ax in x, then the limit of Ay/Ax, as ▲x approaches zero, is called the derivative of y with respect to x. Denoting it by Day (read "the derivative of y with respect to x"), the definition may be expressed by the equation = Ay Dxy lim (1) The results obtained in Section 33, page 94, may now be stated as follows: The derivative of y with respect to x, Day, measures the rate of change of y with respect to x. The derivative of y with respect to x is represented graphically by the slope of a line tangent to the graph of y. That is DxY = m = tan 0. P Ax (2) У The process of finding the derivative of a function is called differentiation, and the succes FIG. 158. -y+Ay sive steps in the process have been given in the section cited above. by Theorem 3 of the preceding section. Applying Theorem 1 in the numerator and Theorems 1 and 2 in the denominator, we get 1. Evaluate the following limits, indicating in detail the use of the theorems in Section 92. (g) y = Vx. Hint. Rationalize the numerator of Ay/Ax before passing to the limit as Ax approaches zero. 3. If f(x) = (x2 + 3x)/(x2 − 1), find lim f(x). Note that the value xa obtained is ƒ(a), provided a ✯ ±1. What happens if a = 4. If f(x) = x2, prove the relations + 1? 5. Prove that the relations in Exercise 4 are true if f(x) is any quadratic function. 6. If f(x) is any function, and Ax = a - x, show that the first relation in Exercise 4 is true if and only if the second is true. What is the graphical significance of each relation? NOTE. A function is said to be continuous at x = a if the first relation in Exercise 4 is true for the given function. It can be proved that the algebraic and transcendental functions studied in this course are continuous for all values of x for which they do not become infinite. If a function becomes infinite as x approaches a, then f(a) has no meaning. Hence x = a is a point of discontinuity. There are other types of points of discontinuity. 7. Prove that (a) any polynomial, (b) any rational function, is continuous for all values of a, except, in (b), for the values for which the function becomes infinite. 8. If u is a function of x, what is the value of lim Au? of lim ? of Au2 lim Ax=0 Ax Ax÷0 = lim Au Au Ax=0 Ax÷0 Ax 94. Fundamental Formulas for Differentiation. The rules in this section are useful in differentiating a function without the labor involved in computing the limit considered in Section 93. Theorem 1. The derivative of a constant is zero; that is, For the graph of y = c is a straight line parallel to the x-axis, whose slope, m = Day D2c, is zero. = y 10 y=x = Dzx, is FIG. 159. Theorem 3. The derivative of a constant times a function is equal to the constant times the derivative of the function. Symbolically, if u is any function of x, What is the graphical interpretation of this theorem, in the light of the Theorem on page 89? Theorem 4. The derivative of the sum of two functions is equal to the sum of their derivatives. Symbolically, if u and v are any two functions of x Corollary. The derivative of the sum of several functions is the sum of their derivatives. Theorem 5. The derivative of the nth power of a function of x is n times the (n - 1)st power of the function times the derivative of the function with respect to x. Symbolically, if u is any function of x Dxun = nun-1 Dxu. (u + Au) n = un + nun−1 Au + (5) n(n − 1) |