INFINITESIMAL CALCULUS 15 and, in general, if y^ . y n we have + J_ % y n dx Agaii), if >j = > we have u = vy, consequently dy dx 1 du v dx du dv -5- - U T dx dx 17. Next, to differentiate a function of a function of x; let y = f(x), and u = ((>(y), to find . Suppose y lt u to be the values which y and u assume when a; becomes x lf then u 1 -u u l ~u y-y Au_Au Ay a i ~ x 2/1 ~y x i ~ x & x ^J ^ x Hence, since, as proved already, the limit of the product of two variable quantities is the product of their limits, we have du _du dy dx dy dx Consequently, the derived function of u with respect to x is the pro- duet of its derived with respect to y and of the derived of y with respect to x. Again, if we suppose u = x, our equations become y=f(x), and x=<j>(y). In the former y is regarded as a function of x, and in the latter x as the corresponding function of y. Such functions are said to be inverse to each other; and in this case we have dx dii dx dy 1 = T i > or T- = 1 +JT dy dx dy dx 18. There exist in analysis a small number of simple or elemen tary functions, each of which requires a special investigation in order to find the corresponding derived function. When these have been established the differentiation of functions composed of these ele mentary functions can be readily obtained, by applying one or more of the principles just established. 19. We commence with the equation y = x", in which ?t is a constant. (1) Let n be an integer, and y^ the value which y assumes when x becomes x l ; then Now the limit of the right hand side when x = x 1 is nx"- 1 ; accord ingly we have in this case dx" = n _ 1 dx (2) Lcty = x", where m and n are integers. Here y n x m , and accordingly ny n - iC ^ = mx m 1 ; hence we get ^^ ^ ff ->> //-> in - - 1 dx n (3) Let y = x *= ; then, from 16, we get -^ = -mx x m dx Consequently we get the following rule, applicable in all cases, for the differentiation of a power of x : Diminish the index by unity, and multiply the power of x thus obtained ly the original index. 20. We shall next consider the elementary circular and trigono metrical functions. Let y = sin x. Then y = sin (a; + h) ; y-i-y _ sin (x + h)- sin x_ 2 . h h -^ h . ^r- h ]3ut _ s i n becomes unity in the limit, and consequently fi 4 dy ~~j == COS X> dx In like manner it is easily seen that d cos x dx d tan a; d dx dx cos g a; + sin 2 a; d sin 35 d cos x cos x ; sin x -, dx dx __ cos 2 a; Similarly ^ = __ <^2. = sec x tan x. dx sin 2 a; dx Corresponding to these trigonometrical functions we have the circular functions, sin - a?, cos - a:, tan~ ! a?, &c. If y = siu- 1 x, we have sj^sin y, and hence dx _ , . di/ 1 ] dy dx cosy /J~ d sin" 1 *; _ 1 dx /i ~yj - 1 d tan ~ l x 1 In like manner d cos"^ _ dx Vl - a; 2 dx "l+a;" 21 . Next, let y = log a x. Here T/J = log u (a; -1- 7; ) ; Let = u ; then x 1 h _ of ,l + *L = J_ lo ga (1 + U} = J- loga (1 + tt) . X J XU X The limiting value of (1 + u} " when u = 0, i. e. , of M + ) when z increases indefinitely, is represented by the letter e (see ALGEBIIA, vol. i. p. 558), and is the base of the natural or Naperian system of logarithms. Hence we have d Og a X 1 , - - = lOQat . dx x If c be taken as the base of our system of logarithms, we have d log x 1 dx x In our subsequent investigations we shall suppose all logarithms, unless otherwise specified, referred to this base, and omit the suffix. 22. The method of differentiation of an exponential function follows immediately from the preceding. For let y=*a x , then logy = x log a, y dx dx We add a few examples for the purpose of showing the appli cation of the preceding results to the differentiation of more com plex functions. (i) y=&. Here log y^x log x ; y dx Hence =(l+log*)a6. dx in , / _i f)f , x W U~ i(J l-y- va^ + x Here y = log x - log (a z + x~) ; dy 1 ffl 2 dx x tl Here y = Hog
= i log dy _ -x = lo -1 (4) Prove that sin 2 * (sin"a* sin nx) = n sin"+ 1 a; sin (n + l)x. dx Here (sin"a; sinwa;) = n sin" - l x (cos a; sin nx + sin a; cos nx) dx n sin"-^ sin (w-fl)a; ; . . &c. _! (5) w = tan 1 = tan y ; from this we get (6) If y = log sin x, prove that -^ = cot x.
