Riesz representation for linear functionals on space of continuously differentiable functions

Question

It is known (Riesz Theorem) that every linear functional $f$ on $X=C[a,b]$ can be represented as $\int_a^bx(t)dv(t)$ for all $x\in X$, where $v$ has bounded variation. Is it true that every linear functional $f$ on $Y=C^1[a,b]$ can be represented as $\int_a^by(t)v(t)dt$ for some continuous $v$?