Page 51 - 12-Math-7 Vectors

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version: 1.1
When y ∈  - π ,0  ,cosec y and cot y are negative
 2 

As cosec y = x, so x is negative in this case
and cot y = - cosec2 y -1 = - x2 -1 when x < -1

-1
x - x2 -1
( )Thu=s ddx Co sec-1 x ( x < -1)

-1

= (-x) x2 -1 ( x < -1)

ddx co sec-1 x  = - | x | 1 for x ∈[-1,1]'
x2 -1

Proof of (5). is left as an exercise
Proof of (6). is similar to that of (4)

Example 1: Find dy if =y x Sin -1 x  + a2 + x2
dx a 

Solution: Given that=y x Sin -1  x  + a2 + x2
 a 

Differentiating w.r.t. x , we have

( ) =dy d x d  x x  d
dx dx a dx a  dx
 x Sin-1 + a=2 + x2  Sin-1 + a2 + x2 1/ 2


1 . Sin-1 x + x. 1 d  x  1 d1 -1
a dx  a  2
( ) ( )= 2 . + . a2 + x2 2 a2 + x2

1 -  x dx
 a

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