Editing Calculus (section)

=== Leibniz notation ===
{{Main|Leibniz's notation}}

A common notation, introduced by Leibniz, for the derivative in the example above is
:<math>
\begin{align}
y&=x^2 \\
\frac{dy}{dx}&=2x.
\end{align}
</math>
In an approach based on limits, the symbol {{math|{{sfrac|''dy''|'' dx''}}}} is to be interpreted not as the quotient of two numbers but as a shorthand for the limit computed above.<ref name=":4" />{{Rp|page=74}} Leibniz, however, did intend it to represent the quotient of two infinitesimally small numbers, {{math|''dy''}} being the infinitesimally small change in {{math|''y''}} caused by an infinitesimally small change {{math|'' dx''}} applied to {{math|''x''}}. We can also think of {{math|{{sfrac|''d''|'' dx''}}}} as a differentiation operator, which takes a function as an input and gives another function, the derivative, as the output. For example:
:<math>
\frac{d}{dx}(x^2)=2x.
</math>

In this usage, the {{math|''dx''}} in the denominator is read as "with respect to {{math|''x''}}".<ref name=":4" />{{Rp|page=79}} Another example of correct notation could be:
:<math>\begin{align}
g(t) &= t^2 + 2t + 4 \\
{d \over dt}g(t) &= 2t + 2
\end{align}
</math>

Even when calculus is developed using limits rather than infinitesimals, it is common to manipulate symbols like {{math|'' dx''}} and {{math|''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]].