DIFFERENTIATION OF TRANSCENDENTAL FUNCTIONS, 16. The transcendental functions may be comprised in two classes; first, the logarithmic, including the exponential; and, secondly, the trigonometrical. I. Logarithmic and Exponential Functions. 17. To find the differentials of u = log . x and u = log.x. u' h) a' = log. (x + 4) = log. { x ( 1 u' .. h h a + h (1+ х h 1 — log. (1 + 2) = 1. log. (1+" ) = log. (1 h х h -1) 1.2 1.2.3 + 1.2.3 and when his continually diminished towards zero, the second member of this tends to become The multiplier rithms whose base is a, and therefore the differential of the logarithm of a quantity is found by dividing the differential of the quantity by the quantity itself, and multiplying the quotient by the modulus of the system. is called the modulus of the system of loga If the base of the system be e, then log, e = 1, and if u Hence the differential of the Napierian logarithm of a quantity is equal to the differential of the quantity divided by the quantity itself. 18. To find the differentials of u = a2 and u = By the principles of logarithms, we have log, u = x log, a, and differentiating both members, we get du શ = dx. log, a, or du = udx. log, a = a dx. log, a; Hence to differentiate a variable power of a constant quantity, multiply that power by the differential of the index, and the result by the Napierian logarithm of the constant quantity. If a = e; then we have log, e = 1, and therefore if u = e*, e, and d u = e* dx. Hence the differential of e" is the product of e, and the differential of the exponent. If we put z = dz = EXAMPLES. 1 log. (a + x) - log, (ax). dx a + x d (a x) a + e; then will u = e', and differentiating these, we get e dx and du = e d z = e" e* dx = e e dx; 19. The student will now be in possession of principles to enable him to differentiate the more complex logarithmic and exponential functions; and whenever the base of the system of logarithms is not indicated, it will be understood that the Napierian system is employed. II. Trigonometrical or Circular Functions. 20. Before proceeding to the differentiation of the trigonometrical or circular functions, it will be necessary to prove the following properties:The limiting ratio of the arc, chord, tangent, and sine of a circular arc is a ratio of equality. Let A P be an arc of a circle whose centre is O, and diameter A B. Draw the tangents AC and PT; then CPT is a right angle, and CT is greater than PT; hence A C is greater than AT and PT together, and the arc A P, which is greater than the chord A P, but less than AT and T P together, is also greater than the chord A P, but less than the tangent AC. Now by similar triangles, O A C and O M P, we have because when P approaches to A, then M also approaches to A. P approaches A, the magnitude of the line BP approaches to that of A B. Taking the limits of the ratios, we get, as in (21), sin (x + h). du dx sin and du = sin x d x. 23. To find the differential of u = tan x. tan x cos x, by the principles of trigonometry, we have or cos x d x = cos x d sec x + sec x d cos x = 0, or cos x d sec x-sec x sin x dx = 0; therefore, by transposing and dividing by cos x, we have Similarly, d sec x = secx tan x dx. d cosec x = - cosec x cot x d x. 25. To find the differential of u = vers x. Since vers x = 1- cos x; therefore d u = d (1 - cos x) = sin x ax; Inverse Trigonometrical Functions. 26. In all these investigations respecting trigonometrical functions, the arc has been considered as the independent variable, and the sine, cosine, tangent, etc., as functions of it. But an arc may also be regarded as a function of its sine, cosine, tangent, etc., and the following very convenient notation, due to Sir John Herschel, is now generally employed to designate such inverse functions. Thus if y sin x, the inverse notation will be x siny, the former expressing the idea that y is equal to the sine of the arc denoted by x, and the latter that x is equal to the arc whose sine is y. In like manner, if u = tan x, then u is the arc whose tangent is equal to x. = = 27. Before proceeding to the differentiation of these inverse trigonometrical functions, it will be necessary to prove the following Relation between Inverse Differential Coefficients. For if y be a function of x, then inversely x will also be a function of y; now if h and k be the simultaneous increments of x and y, then as h approaches 0, k will also approach 0, and by common algebra, Now the limiting value of increment of x, approaches y, the limiting value of x' y' x' y dy or may be found when h, the -x dx 0. In like manner, since x is a function of -x dx y' or may be found when k, the increy dy ment of y, approaches 0, which it does when h approaches 0; hence, taking the limiting value of both ratios, we get dy dx ; then will x Ex. Let y (x + 1) (x + h + 1) x = x + 1 1 (x + 1) (x + h + 1) 1 (x + 1) and y is changed into y + k; x = y+k 1 (y + k) |
