Intuition behind Lagrangian of a optimization problem

Question

Consider the constrained optimization problem below for variable $x$ given as $$\max_{x}f(x) \\ s.t.~g(x)\leq B$$ Now Lagrangian is given as $$L(x)=f(x)-\lambda(g(x)-B)$$Then, taking the derivative and equating to zero gives us the condition that $${\delta f(x) \over \delta x}=\lambda{\delta g(x) \over \delta x}$$Cancelling out $\delta x$ on both sides (does that make sense?) and re-arranging, we get the condition $${\delta f(x) \over \delta g(x)}=\lambda$$What is the intuition behind this condition?