section_path: Uses/Derivative
section_name: Derivative
paragraph_text: Derivative
  
  The derivative is defined formally as a limit. In the scope of real analysis, the derivative is first defined for real functions f {\displaystyle f} defined on a subset E ⊂ R {\displaystyle E\subset \mathbb {R} }. The derivative at x ∈ E {\displaystyle x\in E} is defined as follows. If the limit of f (x + h) − f (x) h {\displaystyle {\frac {f(x+h)-f(x)}{h}}} as h → 0 {\displaystyle h\rightarrow 0} exists, then the derivative at x {\displaystyle x} is this limit.
  
  Equivalently, it is the limit as y → x {\displaystyle y\rightarrow x} of f (y) − f (x) y − x. {\displaystyle {\frac {f(y)-f(x)}{y-x}}.}
  
  If the derivative exists, it is commonly denoted by f ′ (x) {\displaystyle f'(x)}.