Gradient Operator Distributes over Addition

Theorem
Let $\mathbf V$ be a vector space of $n$ dimensions.

Let $\tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ be the standard ordered basis of $\mathbf V$.

Let $\map f {x_1, x_2, \ldots, x_n}, \map g {x_1, x_2, \ldots, x_n}: \mathbf V \to \R$ be differentiable real-valued functions on $\mathbf V$.

Let $\nabla f$ denote the gradient of $f$.

Then:
 * $\nabla \paren {f + g} = \nabla f + \nabla g$