Question

Find the second derivative, do not have to simplify

Answer 1

$f(x)=\sec(\pi x)=\dfrac{1}{\cos(\pi x)}=[\cos(\pi x)]^{-1}\\\\\text{use}\\\\(a^n)'=na^{n-1}\\\\(\cos x)'=-\sin x\\\\f'(g(x))=f'(g(x))\cdot g'(x)$

$f'(x)=\left\{[\cos(\pi x)]^{-1}\right\}'=-[\cos(\pi x)]^{-2}\cdot[-\sin(\pi x)]\cdot\pi\\\\=\dfrac{\pi\sin(\pi x)}{[\cos(\pi x)]^2}=\pi\dfrac{\sin(\pi x)}{\cos(\pi x)}\cdot\dfrac{1}{\cos(\pi x)}\\\\=\pi\tan(\pi x)\sec(\pi x)$