Difference between differentiation and derivatives [closed]

What is the difference between differentiation and derivatives of a function ?

Solution 1:

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.

Solution 2:

Depending on the mathematical school, there is a difference at the definition level. For example Fikhtengol'ts (aka Russian school) (check in preview mode).

• 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.