Writing the Derivative Function

The Language of Mathematicians

In mathematics, just as in language, we sometimes have several ways to say the same thing. For derivatives, there are a few different systems of writing, or notations. The two most famous were created by the giants of calculus, Isaac Newton and Gottfried Wilhelm Leibniz, and we still use their notations today.

Although they look different, they both mean the same thing: indicating the derivative of a function. Understanding both is very helpful, as each notation has its advantages and is often used in different contexts.

Varieties of Derivative Notation

Here are the two most common ways to write a derivative, each with its own style.

Leibniz's Notation

Introduced by Gottfried Wilhelm Leibniz, this notation is very descriptive. If we have a function $y = f(x)$, its derivative is written as:

This is read as "the derivative of y with respect to x" (or "dee-why dee-eks"). Writing $\frac{dy}{dx}$ explicitly tells us that the function $y$ is being differentiated with respect to the variable $x$. This clarity is invaluable, especially in fields like science and engineering where we often work with many interconnected variables, so there is no confusion.

Newton's Notation

This notation, which we often see in textbooks, uses a prime mark ('). If we have a function $y = f(x)$, its derivative is written as:

We read this as "y prime" or "f prime x". This method is very quick and concise, suitable when it is already clear from the context that we are differentiating with respect to the variable $x$. Because of its simplicity, this notation is a favorite for straightforward calculations.

When to Choose Which Notation?

Newton's Notation ($f'(x)$) is great for simple situations. Think of it like a nickname, short and practical for everyday use, especially when everyone already understands which variable is being discussed.

Leibniz's Notation ($\frac{dy}{dx}$) excels in more complex situations. This notation is like a full name with a title, so it's very formal and precise, and it can eliminate any possibility of misinterpretation. This is crucial, for instance, in related rates problems or multivariable calculus, where clarity is key.

Ultimately, both notations are essential tools in our mathematical "toolbox".