A course of pure mathematics by Hardy G. H. (Godfrey Harold) 1877-1947

Author:Hardy, G. H. (Godfrey Harold), 1877-1947 [Hardy, G. H. (Godfrey Harold), 1877-1947]
Language: eng
Format: mobi, pdf
Tags: Calculus, Functions
Publisher: Cambridge, At the University Press
Published: 1921-06-17T16:00:00+00:00

14—2

119. D. Transcendental Functions. We have already proved (Ex. xxxix. 4) that

D x sin x = cos x, D x cos x = — sin x.

By means of Theorems (4) and (5) of § 113, the reader will easily verify that

D x tan x = sec 2 x, D x cot x = — cosec 2 x,

D x sec x = tan x sec x, D x cosec x = — cot x cosec x.

And by means of Theorem (7) we can determine the derivatives of the ordinary inverse trigonometrical functions. The reader should verify the following formulae :

D x arc sin x = + 1/V(1 ~ # 2 )> D x arc cos x = + 1/V(1 — ^e 2 ),

DZ arc tan a? = 1/(1 + x 2 ), D x arc cot # = — 1/(1 + x-) }

D x arc sec x = + 1/{#V(# 2 ~ !)}> A» arc cosec x — + l/{#\/(# 2 — 1)}. In the case of the inverse sine and cosecant the ambiguous sign is the same as that of cos (arc sin x\ in the case of the inverse cosine and secant the same as that of sin (arc cos x).

The more general formulae D x arc sin (as/a) = + l/\/(a 2 — # 2 ), D x arc tan (x/a) = a/(# 2 4- a 2 ),

which are also easily derived from Theorem (7) of § 113, are also of considerable importance. In the first of them the ambiguous sign is the same as that of»a cos {arc sin (x/a)}, since

a VU - O 2 /a 2 )} = ± V<> 2 - x 2 ) according as a is positive or negative.

Finally, by means of Theorem (6) of § 113, we are enabled to differentiate composite functions involving symbols both of alge braical and trigonometrical functionality, and so to write down the derivative of any such function as occurs in the following examples.

Examples XLIV.* 1. Find the derivatives of

cos m #, sin™.*?, cos# m , sin^ m , cos (sin x\ sin (cos x\

cos x sin x

.. 0 - r, - ,„ . „ — r . .V(a 2 cos 2 x + 6 2 sm 2 x) '

x arc sin x + v/( 1 — x 2 ), ( 1 -f ^') arc tan »Jx — Jx.

* In these examples m is a rational number and a, b t ... , a, /3, ... have such values that the functions which involve them are real.

2. Verify by differentiation that arcsiri x -f arc cos x is constant for all values of x between 0 and 1, and arc tano7+arccot# for all positive values of x.

3. Find the derivatives of

arc sin V(l - # 2 ), arc sin (2^ ^(1 - # 2 )}, arc tan (^

\\-ax

How do you explain the simplicity of the results ?

4. Differentiate

1 ax+b 1 ax+b

5. Show that each of the functions Sarcsin J(^\ Cretan J(^\ arc sin VK« -«)(*- ffl}

has the derivative

6 Prove that rf

arc cos

//cos30\) //

VVcos 3 *;/ V'V

^cos 6 cos 3^/ '

(Math.

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