Second derivative of sec^2 (πx)

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)

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!