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dear nikhil , i m explaning this with the help of example ...
suppose we have to diffenertiate sin-1x wrt cos-1x ....
this means , methametically this can be written as dsin-1x/dcos-1x
now , let sin-1x = Z & cos-1x = Y
Z = sin-1x
dZ/dx = d/dx (sin-1x) = 1/(1-x2)1/2 .................1
Y = cos-1x
dY/dx = -1/(1-x2)1/2 .....................2
dividing 1 by 2
dZ/dY = -1
now , Z = sin-1x & Y = cos-1x so
differentiation of sin-1x wrt cos-1x is -1 ...
now see another example ,
differentiate sin-1x wrt tan-1x ...
let Z = sin-1x & Y = tan-1x then
dZ/dx = 1/(1-x2)1/2 & ............1
dY/dx = 1/1+x2 ................2
divide 1 by 2
dZ/dY = 1+x2/(1-x2)1/2 = differentiation of sin-1x wrt tan-1x ...
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