Derivative notation review

Lagrange's notation: $f^{\prime }$‍

Leibniz's notation: ${\displaystyle \frac{dy}{dx}}$‍

Newton's notation: $\dot{y}$‍

Derivatives are the result of performing a differentiation process upon a function or an expression. Derivative notation is the way we express derivatives mathematically. This is in contrast to natural language where we can simply say "the derivative of...".

In Lagrange's notation, the derivative of $f$‍ is expressed as $f^{\prime }$‍ (pronounced "f prime" ).

This notation is probably the most common when dealing with functions with a single variable.

If, instead of a function, we have an equation like $y=f(x)$‍, we can also write $y^{\prime }$‍ to represent the derivative. This, however, is less common to do.

In Leibniz's notation, the derivative of $f$‍ is expressed as ${\displaystyle \frac{d}{dx}}f(x)$‍. When we have an equation $y=f(x)$‍ we can express the derivative as ${\displaystyle \frac{dy}{dx}}$‍.

Here, ${\displaystyle \frac{d}{dx}}$‍ serves as an operator that indicates a differentiation with respect to $x$‍. This notation also allows us to directly express the derivative of an expression without using a function or a dependent variable. For example, the derivative of $x^2$‍ can be expressed as ${\displaystyle \frac{d}{dx}}(x^2)$‍.

This notation, while less comfortable than Lagrange's notation, becomes very useful when dealing with integral calculus, differential equations, and multivariable calculus.

In Newton's notation, the derivative of $f$‍ is expressed as $\dot{f}$‍ and the derivative of $y=f(x)$‍ is expressed as $\dot{y}$‍.

This notation is mostly common in Physics and other sciences where calculus is applied in a real-world context.