A new chain rule formula is shown for the distributional derivative of the composite function \(v(x)=B(x,u(x))\), where \(u: (a,b) \to\mathbb R^n\) has bounded variation, \(B(x,.)\) is continuously differentiable and \(B(., u)\) has bounded variation. Applications of this formula are shown in order to deal with the discontinuous flux appearing in conservation laws in one space variable.