Lagrange's notation: ff'
Leibniz's notation: dydx\dfrac{dy}{dx}
Newton's notation: y˙\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 ff is expressed as ff' (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)y=f(x), we can also write yy' to represent the derivative. This, however, is less common to do.

Leibniz's notation

In Leibniz's notation, the derivative of ff is expressed as ddxf(x)\dfrac{d}{dx}f(x). When we have an equation y=f(x)y=f(x) we can express the derivative as dydx\dfrac{dy}{dx}.
Here, ddx\dfrac{d}{dx} serves as an operator that indicates a differentiation with respect to xx. 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 x2x^2 can be expressed as ddx(x2)\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 ff is expressed as f˙\dot f and the derivative of y=f(x)y=f(x) is expressed as y˙\dot y.
This notation is mostly common in Physics and other sciences where calculus is applied in a real-world context.

Check your understanding

Problem 1
g(x)=xg(x)=\sqrt x
How can we express the derivative of x\sqrt x?
Choose all answers that apply:
Choose all answers that apply: