angle be 84°, and let it receive small increments of From this it will be seen that the ratio of the rate of variation of the function to the rate of variation of the variable 0001593 *0000175 =91 028 etc. But sec 84° x tan 84°= 9.5667722 × 9·5143645 =91 0217475. Therefore the required ratio =sec 84° x tan 84° (approximately), the error occurring in the third decimal place. And this result is of the general form differential co-efficient of sec x=sec x tan x. Similar results may be found for the cosine, cotangent, and cosecant of an angle. XVII. The Differential Co-efficient of a Logarithm. 74. Assuming the exponential theorem we may show that where M is the modulus and is found to be '43429. For instance, take log (1+315). 34 Now the following will be found in any tables of logarithms— log 41713-4.6202714, If we take the differences of these, we obtain *0000104, Thus, if 41713 receive small increments of 1, the function receives increments of 0000104; i.e., the ratio of the rate of variation of the function to the rate of variation of the variable = '0000104: 1; or, the differential co-efficient of log 41713=·0000104. Now the general form is differential co-efficient of log x= 1 and the above result does not, at first sight, appear to be of this form. We shall see, presently, that it is. Converting the above into Naperian logarithms, we have Taking the difference, as before, between (1) and (2), we obtain •000024 1 41713 Therefore, taking Naperian logarithms, we have, the ratio of the rate of variation of the function to the rate of variation of the variable=000024 : 1 1 = 41713 or, differential co-efficient of Nap. log 41713= 1 41713 75. We may now show that the result obtained in the previous article is of the same form. (Taking logs to base 10), log 41713=46202714, 104 6202714-41713; 104 6202818=104 6202714+0000104 that is, again, 76. The following considerations are of the utmost importance, as they embody the whole principle, not only of differentiation, but also of successive differentia tion. It must be remembered that when a body is moving, not uniformly, but with accelerated motion, its rate at any instant is not represented by the space it would pass over in the next unit of time, but by the space it would pass over if it moved uniformly, with the velocity it had at that instant, for the next unit of time. Let QA be a cube, which has been growing to its present size, and let OA=x=OB=OC, ≈ being the variable on which size of cube depends, and let OA receive, in the ordinary course of its increase, an increment Aa, and let Bb, Cc be the corresponding increments in OB, OC. The edge of the cube (x) has, then, received a certain increase-i.e., its rate of increase at the instant it has become x is represented by Aa or Bb or Cc, in the three respective directions. Our aim is to find the ratio of the rate of increase of the cube, to this rate of increase of x-i.e., the differential co-efficient of x; we have therefore to find the corresponding rate of increase of the cube to the increase of its edge (x). Now if the cube had been stopped suddenly on its increasing course at the instant at which we found it in the form of QA, its rate of increase in the directions of Aa or Pd', Bb or Pd", Cc or Pd", corresponding to this rate of increase of x, would be represented by the figures Pa, Pb, and Pc, for the face of the cube would have to remain of the same size as we found it at the instant, in order that we may satisfy the condition of uniformity, already alluded to, in calculating the rate. Therefore the first rate of increase of cube = = Pa+Pb+ Pc = Pd' x face of cube + Pd" × face of cube = face of cube x (Pd'+Pd" + Pď") face of cube x 3Aα = face of cube x 3 (rate of increase of x). Therefore rate of increase of cube =3x (face of cube), rate of increase of x rate of increase of x3 i.e., = 3x2, rate of increase of x or differential co-efficient of x3-3x2; and this is the first differential co-efficient of a3. 77. Now, a moment ago, we suddenly stopped the cube in its growth. If we had not, it would have increased in size, and, as a necessary and obvious consequence, its three faces would have increased in area. (A cube of course has six faces, but there are only now three under consideration, since the cube is not sup |