P'N PP' sin2POM COS POM sin' POM = cosec POM.cot POM; or, the differential co-efficient of cosec POM - cosec POM.cot POM. = And this is of the general form— differential co-efficient of cosecx= cosec x cot x. XIV. The Differential Co-efficients of the Inverse 67. In a similar manner the differential co-efficients of the inverse trigonometrical functions may be obtained. Sin-1 means the angle whose sine is x; let this angle be POM. Then sin-1x is the function and sin x the independent variable; and the ratio of the rate of variation of the angle (i.e., the arc) to the rate of variation of the sine=PP' : P'N or, the differential co-efficient of sin-1=x 1 68. Now let the function be tan-1x. We found before that the ratio of the rate of variation of the tangent to the rate of variation of the angle was sec2 POM. Therefore the ratio of the rate of variation of the angle to the rate of variation of the tangent if POM be the angle whose tangent is x; or, the differential co-efficient of tan-1=; 1 1 + x2 69. Again, let the function be sec-1x. We found that the ratio of the rate of variation of the secant to the rate of variation of the angle was sec POM.tan POM. Therefore the ratio of the rate of variation of the angle to the rate of variation of the secant = 1 sec POM. tan POM 1 sec POM sec2POM – 1 1 = x√x2-1 or, the differential co-efficient of sec-1x= if POM be the angle whose secant is x. 1 x √x2 - 1 Similarly we may find the differential co-efficients of the other inverse trigonometrical ratios. XV. The Value of xo. 70. It might appear to some that it would be sufficient to say that a quantity which is continually diminishing may be made as small as we please, without the proviso that it may be made smaller than any assignable quantity. But on closer inspection it will be found that, in some cases, quantities may be continually diminishing and yet never become smaller than a certain quantity, which is then the ultimate value, or limit, when the decrease has been carried out to an indefinite extent. For instance, suppose we take the number 100, and take its square root; this will be 10. Now take the square root of 10; this will be 3 followed by a decimal. Take the square root of this, and the result will be 1 followed by a smaller decimal, and so on. However many times we take the square root the 1 will always remain, though the decimal part may be made smaller than any assignable quantity. The limit, then, of any number, when the square root has been taken an infinite number of times, is 1. Let x be any number; then the square root of x is written, and the square root of this again is x1, and when we have taken the square root n times the result 1 will be 2; and when we have taken the square root an infinite number of times, i.e., when n has become ∞, the result is x or xo, and therefore x = 1. 1 XVI. The Differential Co-efficients of the Trigonometrical Functions (Arithmetically). 71. Consider the following :— Here the first column gives the successive values of an angle of 70° as it receives small increments of 0001 degrees. The second column gives the corresponding arcs or circular measure of these angles. The third column the differences of these arcs, or the rate of variation of the arcs. The fourth column gives the natural sines of the angles, which may be found in any book of logarithmic tables. The fifth column gives the differences of these. Here the sine is the function of the arc; and the rate of variation of the function is given by the fifth column. Therefore the ratio of the rate of variation of the function to the rate of variation of the variable =cosine of an angle whose circular measure is 1.221730 (approximately) = cos 70°, or, the differential co-efficient of sin 70°= cos 70°. And this is of the general form differential co-efficient of sin x cos x. The error committed in the above is considerable, because the tables are only carried to 7 places of decimals. Now, the smaller the increments are, the more true is the result, and for very small increments it would be necessary to have tables calculated to a far greater number of decimal places. In the following example the increment is comparatively large— Therefore, the ratio of the rate of variation of the function to the rate of variation of the variable ⚫0015114 = *001745 ='866 etc. =cos 30° (approximately), and, therefore, the differential co-efficient of sin 30° is cos 30°, and this is of the general form differential co-efficient of sin x cos x. Similarly, the differential co-efficient of the cosine may be shown to be of the general form, from the actual numbers. 72. Now let us take the tangent, and suppose the angle to be 14° and let it receive small increments of •1°. Then Here the ratio of the rate of variation of the tangent (function) to the rate of variation of the arc (variable) Therefore the required ratio=sec214° (approximately), the error occurring in the fourth decimal place. Therefore, approximately, differential co-efficient of tan 14° = sec214°, and a similar result may be obtained for any other angle. Further, it will be seen that this result is of the general form differential co-efficient of tan x= =sec2x. 73. Now take the secant as the function, and let the |