Question

secx -118.secx1

Answer 1

Let the given function be,

f( x )= secx−1 secx+1 = 1 cosx −1 1 cosx +1 = 1−cosx 1+cosx

Differentiate the function with respect to x,

f ′ ( x )= ( 1+cosx ) d dx ( 1−cosx )−( 1−cosx ) d dx ( 1+cosx ) ( 1+cosx ) 2 = ( 1+cosx )( sinx )−( 1−cosx )( −sinx ) ( 1+cosx ) 2 = sinx+sinxcosx+sinx−sinxcosx ( 1+cosx ) 2 = 2sinx ( 1+cosx ) 2

Simplify further,

f ′ ( x )= 2sinx ( 1+ 1 secx ) 2 = 2sinx sec 2 x ( 1+secx ) 2 = 2sinx( 1 cos 2 x ) ( 1+secx ) 2 = 2tanxsecx ( 1+secx ) 2

Therefore, the derivative of the given function is 2tanxsecx ( 1+secx ) 2 .