Question

Second derivative of sec^2(πx)

Answer 1

First derivative  is  2 pi tan(pi x) sec^2 (pi x)

second derivative  looks really messy!

using the Product rule :-

=  2 pi tan(pi x) * 2pi tan (pi x) sec^2 (pix)+sec^2(pi x) *  2 pi^2 sec*2 ( pi x)

=  4 pi^2 tan^2 (pi x) sec^2 (pi x) + 2 pi^2 sec^4 (pi x)

=   2 pi^2 sec^2 (pi x) [ 2 tan^2 (pi x) + sec^2(pi x)]


Phew!

Answer 2

 let y = sec^2 ( pi x ) 

y' = 2 sec ( pi x ) sec( pi x ) tan ( pi x ) pi 

y' = 2pi sec^2 ( pi x ) tan ( pi x ) 

y''= 2pi sec^2 ( pi x ) * sec^2 ( pi x ) * pi + 2pi tan ( pi x ) * 2pi sec^2 ( pi x ) tan ( pi x ) 

y'' = 2 pi^2 sec^4 ( pi x ) + 4 pi^2 sec^2 ( pi x ) tan^2 ( pi x )