help about supremum norm proof

Question

T:( $C'[0,1] ,||x||₁)\to (C[0,1]$,||x||∞)

Tx(t)=x'(t) for any x is in C'[0,1]

$||x||₁=||x||_{\infty}+||x′||$∞ how can we prove that equality?

$||x||_1 = \sum_{n=1}^{\infty}|x_{n}|$

$||x||_{\infty}=\sup|x_n|$