Difference between differentiation and derivatives What is the difference between differentiation  and derivatives of a function ?
 A: Differentiation is a process that gives you the derivative. Or, symbolically, if $f$ is a differentiable function, then $f'$ is its derivative and the map $f \to f'$ is differentiation.
A: Depending on the mathematical school, there is a difference at the definition level. For example Fikhtengol'ts (aka Russian school) (check in preview mode). 


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*Page 145. Function $f$ is said to have derrivative in $x_0$ if the following limit exists:
$$\lim\limits_{x\rightarrow x_0} \frac{f(x)-f(x_0)}{x-x_0} \text{ or } \lim\limits_{h\rightarrow 0} \frac{f(x_0+h)-f(x_0)}{h}$$ this limit is also called derivative of $f$ in $x_0$.

*Page 165. Function $f$ is said to be differentiable in $x_0$ if its increase $f(x_0 +\Delta x)-f(x_0)$ can be expressed as
$$f(x_0 +\Delta x)-f(x_0)=A\cdot \Delta x + o(\Delta x)$$
where A does not depend on $\Delta x$ and $\lim\limits_{\Delta x \rightarrow 0}\frac{o(\Delta x)}{\Delta x}=0$.


Despite these 2 definitions, for functions in one variable, there is a theorem stating that both definitions are equivalent. The difference becomes more obvious when applied to multi-variable functions or functions in various vector spaces.
