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## AP®︎/College Calculus AB

### Unit 2: Lesson 1

Defining average and instantaneous rates of change at a point

# Derivative notation review

Review the different common ways of writing derivatives.
Lagrange's notation: f, prime
Leibniz's notation: start fraction, d, y, divided by, d, x, end fraction
Newton's notation: y, with, \dot, on top

## 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, 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, equals, f, left parenthesis, x, right parenthesis, we can also write y, prime 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 start fraction, d, divided by, d, x, end fraction, f, left parenthesis, x, right parenthesis. When we have an equation y, equals, f, left parenthesis, x, right parenthesis we can express the derivative as start fraction, d, y, divided by, d, x, end fraction.
Here, start fraction, d, divided by, d, x, end fraction 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, squared can be expressed as start fraction, d, divided by, d, x, end fraction, left parenthesis, x, squared, right parenthesis.
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 f, with, \dot, on top and the derivative of y, equals, f, left parenthesis, x, right parenthesis is expressed as y, with, \dot, on top.
This notation is mostly common in Physics and other sciences where calculus is applied in a real-world context.