Gradient of an integration functional

Question

I am trying to derive the formula for the Lagrangian formalism on constraints directly from Lagrangian multipliers, where we try to extremize $S: C^\alpha(\mathbb{R}^n)\to\mathbb{R}, \ S[x(t)]=\int\limits_{t_1}^{t_2} L(x,\dot x,t)\ dt$ on given constraints $g: \mathbb{R}^n\to\mathbb{R}^m$, which leads us to: $$\nabla S=\displaystyle\sum\lambda_i\ (\nabla g)_i$$ But I can't figure out how to compute the gradient of an integration functional, can someone please explain how to do this?