Editing Leibniz's notation (section)

==Use in various formulas==
One reason that Leibniz's notations in calculus have endured so long is that they permit the easy recall of the appropriate formulas used for differentiation and integration. For instance, the [[chain rule]]—suppose that the function {{mvar|g}} is differentiable at {{mvar|x}} and {{math|1=''y'' = ''f''(''u'')}} is differentiable at {{math|1=''u'' = ''g''(''x'')}}. Then the composite function {{math|1=''y'' = ''f''(''g''(''x''))}} is differentiable at {{mvar|x}} and its derivative can be expressed in Leibniz notation as,<ref>{{harvnb|Briggs|Cochran|2010|loc=p. 176}}</ref>
:<math>\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}.</math>
This can be generalized to deal with the composites of several appropriately defined and related functions, {{math|''u''<sub>1</sub>, ''u''<sub>2</sub>, ..., ''u''<sub>''n''</sub>}} and would be expressed as,

:<math>\frac{dy}{dx} = \frac{dy}{du_1} \cdot \frac{du_1}{du_2} \cdot \frac{du_2}{du_3}\cdots \frac{du_n}{dx}.</math>

Also, the [[integration by substitution]] formula may be expressed by<ref>{{harvnb|Swokowski|1983|loc=p. 257}}</ref>

:<math>\int y \, dx = \int y \frac{dx}{du} \, du,</math>

where {{mvar|x}} is thought of as a function of a new variable {{mvar|u}} and the function {{mvar|y}} on the left is expressed in terms of {{mvar|x}} while on the right it is expressed in terms of {{mvar|u}}.

If {{math|1=''y'' = ''f''(''x'')}} where {{mvar|f}} is a differentiable function that is [[Inverse function|invertible]], the derivative of the inverse function, if it exists, can be given by,<ref>{{harvnb|Swokowski|1983|loc=p. 369}}</ref>
:<math>\frac{dx}{dy} = \frac{1}{\left( \frac{dy}{dx} \right)},</math>
where the parentheses are added to emphasize the fact that the derivative is not a fraction.

However, when solving differential equations, it is easy to think of the {{math|''dy''}}s and {{math|''dx''}}s as separable. One of the simplest types of [[differential equation]]s is<ref>{{harvnb|Swokowski|1983|loc=p. 895}}</ref> 
:<math>M(x) + N(y) \frac{dy}{dx} = 0,</math>
where {{mvar|M}} and {{mvar|N}} are continuous functions. Solving (implicitly) such an equation can be done by examining the equation in its [[differential form]],
:<math>M(x) dx + N(y) dy = 0</math>
and integrating to obtain
:<math>\int M(x) \, dx + \int N(y) \, dy = C.</math>
Rewriting, when possible, a differential equation into this form and applying the above argument is known as the ''separation of variables'' technique for solving such equations.

In each of these instances the Leibniz notation for a derivative appears to act like a fraction, even though, in its modern interpretation, it isn't one.