Norm of linear functional in $W^{1,2}$

Question

I want to find the norm of the following linear functional:

$\phi:W^{1,2}_{[0,1]} \rightarrow\mathbb{R}$

$\phi(f)=\int^1_0t\cdot f(t)+ f^{'}(t) dt $

The norm is: $||f||_{W^{1,2}}=||f||_2+||f^{'}||_2$

I didn't get too far:

$|\phi|=|\int^1_0t\cdot f(t)+ f^{'}(t) dt|\leq\int^1_0 |tf| +\int^1_0 |f^{'}|$

And I should get something like this:

$||\phi||\leq L(||f||_2+||f^{'}||_2 )$

Thanks in advance!