section_path: Principles/Leibniz notation
section_name: Leibniz notation
paragraph_text: Leibniz notation
  
  A common notation, introduced by Leibniz, for the derivative in the example above is
  
  In an approach based on limits, the symbol dy / dx is to be interpreted not as the quotient of two numbers but as a shorthand for the limit computed above. Leibniz, however, did intend it to represent the quotient of two infinitesimally small numbers, dy being the infinitesimally small change in y caused by an infinitesimally small change dx applied to x. We can also think of d / dx as a differentiation operator, which takes a function as an input and gives another function, the derivative, as the output. For example:
  
  In this usage, the dx in the denominator is read as "with respect to x ". Another example of correct notation could be:
  
  Even when calculus is developed using limits rather than infinitesimals, it is common to manipulate symbols like dx and dy as if they were real numbers; although it is possible to avoid such manipulations, they are sometimes notationally convenient in expressing operations such as the total derivative.