I don’t understand the definition given in my notebook from my teacher.
Given $f: (a, b) \to \mathbb{R}$ and $x \in (a, b)$. We say $f$ is differentiable at $x$ if there exists $A \in\mathbb{R}$ and a function $\phi : \mathbb{R} \to \mathbb{R}$ such that: $$f(x + h) = f(x) + Ah + |h|\phi(h), \forall h \in (-r, r),$$
Where $r > 0$ is small enough that $(x - r, x + r) \subset (a, b)$ and $\lim_{h \to 0}{\phi(h)} = 0.$
I can't find any document about this so I don't know how it works. Can someone explain more about this definition or give me a document about it.
This definition of derivative is given from the point of view of linear approximation. Namely, if a function $f$ is differentiable at $x_0$, then near $x_0$, $f(x)$ can be approximated by the linear function $f(x_0)+A(x-x_0)$, where $A$ is the slope of the tangent line to the graph of $f$ at $(x_0, f(x_0))$, and the error is a higher order term in terms of $|x-x_0|$.
It is easy to prove that the two definitions are equivalent. Indeed, from the linear approximation formula, it is readily to check that $f'(x_0)=A$. On the other hand, if $f'$ exists at $x_0$, then the difference quotient $$\frac{f(x_0+h)-f(x_0)}{h}$$ has a limit as $h\rightarrow 0$. Let's say the limit is $A$. Hence equivalently, as $h\rightarrow 0$, $$\phi(h):=\frac{f(x_0+h)-f(x_0)}{h}-A\rightarrow 0.$$ Hence $$f(x_0+h)=f(x_0)+h(\phi(h)+A)=f(x_0)+Ah+h\phi(h),$$with $\lim_{h\rightarrow 0}\phi(h)=0,$ as you desired.
Both definitions have advantanges. The limit of difference quotient is from the physical point of view of change of rate, and it also has geometrical meaning as slope of the tangent line at the given point. The linear approximation definition has the advantage that $dy=f'(x)dx$, which promotes the notion of differential.