# Derivative notation review

Review the different common ways of writing derivatives.
Lagrange's notation: $f'$
Leibniz's notation: $\dfrac{dy}{dx}$
Newton's notation: $\dot y$

## What is derivative notation?

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...".

## Lagrange's notation

In Lagrange's notation, the derivative of $f$ is expressed as $f'$ (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 $y'$ to represent the derivative. This, however, is less common to do.

## Leibniz's notation

In Leibniz's notation, the derivative of $f$ is expressed as $\dfrac{d}{dx}f(x)$. When we have an equation $y=f(x)$ we can express the derivative as $\dfrac{dy}{dx}$.
Here, $\dfrac{d}{dx}$ 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 $\dfrac{d}{dx}(x^2)$.
This notation, while less comfortable than Lagrange's notation, becomes very useful when dealing with integral calculus, differential equations, and multivariable calculus.

## Newton's notation

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.

$g(x)=\sqrt x$
How can we express the derivative of $\sqrt x$?