we have the taylor approximation as follows
$$f(x) = f(\bar x) + \nabla f(\bar x)(x − \bar x) + o(\|x − \bar x\|)$$
How to prove this?
If f is continuous over $[\bar x, x]$ and differentiable over $(\bar x, x)$, then:
$$f(x) = f(\bar x) + \nabla f(z)(x − \bar x), \; \text{for some} \; z \in [\bar x, x]$$