By Brian H. Chirgwin, Charles Plumpton

ISBN-10: 0080063888

ISBN-13: 9780080063881

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4. Differentiation of a function of a function. If y = F(u) where u = f(x), clearly y is a function of x. , t h e function F of t h e function / . Examples, (i) (a:4 + a4)6 is the function ue of the function u = a:4 -f a4· (ii) cos (a:3) is the function cos % of the function u = x3. (iii) (cos a;)3 or cos3a; is the function us of the function u = cos x. Suppose t h a t , corresponding t o an increment ox in x, t h e increments in u a n d y are δ% a n d by respectively. Then, assuming t h a t bu -> 0 when <5# -> 0 , t h e i d e n t i t y 02/ by bu bx bu bx gives ■ / = h m - / - = lim l·^-.

6). 46 A C O U R S E OF M A T H E M A T I C S Examples. cos 5x — cos 5 a; — 10 cos3a; sin 2 a; + 5 cos x sin 4 a;. (i) . · . cosh 5x — cosh 5 a: —- 10 cosh 3 a;(— 1) sinh 2 a; + 5 cosh x(— l ) 2 sinh 4 a;· . · . cosh 5x = cosh 5 a; + 10 cosh 3 a: sinh 2 a; + 5 cosh x sinh 4 a;. sin 5x = 5 sin x — 20 sin 3 a; + 16 sin 5 a;. ·. sinh 5x = 5 sinh x — 20(— 1) sinh 3 a; + 16(— l ) 2 sinh 5 a:. ·. sinh 5x = 5 sinh x + 20 sinh 3 a; + 16 sinh 5 a;. The derivatives of the hyperbolic functions follow at once from the definitions, viz.

D —— i"cos3a;] = —— · —— (cos x) = 3u2 · (— 1) sin x = —3 sin x cos2#. dx du dx The chain rule m a y be generalised. For, if y is a function of u and u is a function of v which is a function of x, t h e n Ay _ dx ày du du dv dv dx This result m a y be extended t o a n y n u m b e r of functions. Example. d d — (1 + 5 sin3 2a;)4 = 4 ( 1 + 5 sin3 2a:)3 — (5 sin3 2x) ax dx d = 20(1 + 5 sin3 2a;)3 · 3 sin2 2a; -j— (sin 2x) dx = 120 sin2 2a; cos 2a; (1 + 5 sin3 2a;)3. 5. Differentiation of parametric equations.

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A Course of Mathematics for Engineers and Scientists. Volume 1 by Brian H. Chirgwin, Charles Plumpton


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