Error estimate for a Finite Element Method

Question

Let $0=x_0<x_1<...<x_n$ be a partition of the interval [0,1] and let $S=\{v\in C^0([0,1]):v|_{[x_i,x_{i-1}]} \mathrm{is\ a\ linear\ polynomial}, i=1,...,n\ \mathrm{and}\ v(0)=0\}.$Suppose $u(x)\in S$ is such that $$ \int_0^1u'(x)v'(x)dx=\int_0^1f(x)v(x)dx $$ and let $\tilde{u}(x)\in S$ be such that $$ \int_0^1\tilde{u}'(x)v'(x)dx=\sum_{i=0}^N\frac{h_i+h_{i+1}}{2}f(x_i)v(x_i)$$ where $$ h_i=x_i-x_{i-1},\ h_0=h_n=0. $$ I want to show that $$ ||u'-\tilde{u}'||_{L_2}\leq Ch^2(||f'||_{L_2}+||f''||_{L_2}). $$ I've attempted to integrate by parts, use Holder's inequality, and use Poincare's inequality where appropriate, but have come up empty. Does anyone have any suggestions on how to prove this estimate?