By Brian H. Chirgwin, Charles Plumpton

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R. to x : 2 1. ex*. 2. 7. t a n h x eax x . 3. e 8 l n i C . — t a n h 2 x. o 12. log [a; +]/(*2 ~ 1)]. cosh x + sinh x cosh x — sinh a; 16. log 19. log 15. x ]/(a;2 + 1) (*-4) V ( 2 a ; - 3) 5. sinh n a; cosh m a;. 8. log sin x. 9. log cos a;. 11. l o g i i - ^ g - j . 14. 4. eaxsinbx. * . 20. 10. log (a;3 + 1 ) Ï (a;4 - 1)τ 18. (e*-l) (e* + 1) 1 +x\ 1 — x)' 13. V» + log (l - V « ) . cosh a; + cos x sinh x + sin a; 17. t a n ( a + bx). 6. x*l°8x. § 2:5 THE T E C H N I Q U E OF D I F F E R E N T I A T I O N 47 21.

W e express 2/ explicitly in t e r m s of x in t h e form y = s i n - 1 a; t o be read as "sine t o t h e minus one x" or t h e "inverse sine of x;" this m e a n s t h e angle whose sine is x. T h e notation arc sin x is used b y some writers. W e define cos" 1 x, t a n - 1 x, c o t - 1 x, s e c - 1 x a n d cosec" 1 x similarly. The graphs of s i n - 1 x, c o s - 1 x a n d t a n - 1 x are shown in Figs. 11(a), (b), (c). Clearly y = s i n - 1 a? , s i n - 1 (^) = ±ηπ + (— l)n π / 6 where n is 0 or an integer.

On considering f(x, y) it is clear that x and y are the independent variables and therefore in finding the derivative df/dy it is the independent variable x which is kept constant ; there is no need to denote this in any special way. However, in some applications of partial differentiation, notably in thermodynamics, it is not always clear which variables of a group, such as p, v, T, φ, u, are the indepen dent and which the dependent variables. When the context does not make this clear a suffix is put after the brackets enclosing a derivative to indicate the other independent variable.