. An elementary course of infinitesimal calculus . — |^tanh x, Dy = sech* x. 6. 2/ = cosh X cos X iJy = 2 sinh a; cos x. + sinh a; sin x, 7. y = cosh a; sin x Dy = 2 cosh x cos a;. + sinh X cos x, cosh X - cos a; _ 2 sin x sinh a; 8. « = ^r-T ^ , Dy = 7-;—; : r; . ° smh X + sm a; (sinh a; + sin a;) 41. Differentiation of Inverse Functions. If 2/ be a continuous function of .t, then tinder a certaincondition (see Art. 20), which is fulfilled in the case of mostordinary mathematical functions, x will be a continuousfunction of y. If Sa;, %y be corresponding increments of x and y, wehave Sy Sa;_
. An elementary course of infinitesimal calculus . — |^tanh x, Dy = sech* x. 6. 2/ = cosh X cos X iJy = 2 sinh a; cos x. + sinh a; sin x, 7. y = cosh a; sin x Dy = 2 cosh x cos a;. + sinh X cos x, cosh X - cos a; _ 2 sin x sinh a; 8. « = ^r-T ^ , Dy = 7-;—; : r; . ° smh X + sm a; (sinh a; + sin a;) 41. Differentiation of Inverse Functions. If 2/ be a continuous function of .t, then tinder a certaincondition (see Art. 20), which is fulfilled in the case of mostordinary mathematical functions, x will be a continuousfunction of y. If Sa;, %y be corresponding increments of x and y, wehave Sy Sa;_ Bx Sy identically. Hence, since the limit of the product is equalto the product of the limits, dydx^^ dxdy Hence, it being presupposed that yisa. differentiable functionof X, it follows that x is in general a differentiable functionof y, and that the two <lerived functions are reciprocals. The geometrical meaning of this is that the tangent to a curvemakes complementary angles with the axes of x and y. 86 INFINITESIMAL CALCULUS. [CH. II. Fig. 28. 41] DERIVKT) FUNCTIONS. 87 The following cases are important: P. If y=sin-^x (2), we have a; = sin y, -t- = cos y. Hence ^^J—^+ ^ (3). dx cos 3/ V(l — ^^) 2°. If y=cos-a; (4), , dx we have a; = cos y, t- = — sm y, and therefore -^ = —; = + -^ -r (5). dx sin y v(l — x^) The ambiguity of sign in these results is to be accounted foras follows. We have seen that if y = sin^ x, then y is a many-valued function of x; viz. for any assigned value of x (betweenthe limits + 1) there is a series of values of y, and for some ofthese dyjdx is positive, for others negative. See Fig. for cos~a;. If, when x is positive, we agree to understand by sin x theangle between 0 and Jir whose sine is x, we must write i^^^^=+V(I^ ^^^- Similarly if, x being positive, cos- x be restricted to lie between0 and Jtt, we have d , \ /_, dx^ ^-W^) ^^? 3°- If 2/ = tan-a; (8), we have x = tan y, -v-= sec^ y, and therefore -^ = —;;~ = ^
Size: 997px × 2507px
Photo credit: © Reading Room 2020 / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., bookcentury1900, bookdecade1900, bookpublishercambr, bookyear1902
