Calculus

# Leibniz’s Notation & dy/dx Meaning

## Rachel McLean

Subject Matter Expert

Leibniz’s notation is a fundamental type of notation for derivatives. In this article, we’ll discuss the meaning of dy/dx, how to use Leibniz’s notation, and practice some examples.

## Who Is Leibniz?

Gottfried Wilhelm Leibniz (1646 - 1716) was a 17th century German mathematician. He’s often credited with developing many of the main principles of differential and integral calculus, and is primarily recognized for what we now call Leibniz’s notation.

## What Is Leibniz’s Notation System?

Derivative notations are used to express the derivative of a function based on today’s standard definition of a derivative. The instantaneous rate of change, or derivative, of a function $f$ at $x$ is given by: $\frac{d}{{dx}}f\left( x \right) = \mathop {\lim }\limits_{\Delta{x} \to 0} \frac{\Delta{y}}{\Delta{x}} = \mathop {\lim }\limits_{\Delta{x} \to 0} \frac{{f\left( {x + \Delta{x} } \right) - f\left( x \right)}}{\Delta{x} }$

There are many different derivative notations, but Leibniz’s notation remains one of the most popular. Given a function $f$ defined by $y = f(x)$, Leibniz's notation expresses the derivative of $f$ at $x$ as:

$\frac{dy}{dx}$

It might be tempting to think of $\frac{dy}{dx}$as a fraction. In fact, Leibniz himself first conceptualized $\frac{dy}{dx}$as the quotient of an infinitely small change in y by an infinitely small change in x, called infinitesimals. However, this understanding of Leibniz’s notation lost popularity in the 19th-century when infinitesimals were considered too imprecise to define the infrastructure of calculus. Leibniz’s original understanding of $\frac{dy}{dx}$as a quotient has been reinterpreted to align with the modern limit-based definition of a derivative. Now, $dy$and $dx$are generally referred to as differentials instead of infinitesimals. While Leibniz’s notation behaves in a way similar to a fraction, it’s important that you understand the difference.

## What Is dy/dx?

We can think of $\frac{d}{dx}$as an operator defined by the standard limit-based definition of a derivative. Suppose we have a function $f$defined by $y = f(x)$. When we apply the operator $\frac{d}{dx}$to $y$, we have the expression $\frac{d}{dx}y$, or $\frac{dy}{dx}$. This expression represents the derivative of $y$with respect to $x$(note that $y$is the function value of $f$at $x$). In this way, the operator $\frac{d}{dx}$takes in one function, and outputs another! More specifically, the operator $\frac{d}{dx}$ acts on a function to produce that function’s derivative.

Let’s review some examples where Leibniz’s notation is often utilized. Consider the Chain Rule, which helps us differentiate composite functions.

Suppose we have two differentiable functions $f$and $g$, and suppose that $g$is differentiable at $x$and $f$is differentiable at $g(x)$.

Then, the composite function $h = f \circ g$, such that $h(x) = f(g(x))$for all $x$, is differentiable at $x$.

With these conditions satisfied, the Chain Rule states:

$h’(x) = f’(g(x))g’(x)$

Suppose $y = f(u)$and $u = g(x)$. Then, $g’(x) = \frac{du}{dx}$and $f’(u) = \frac{dy}{du}$.

We can translate the above Chain Rule into Leibniz’s notation by writing:

Next, let’s look at the Inverse Function theorem.

Suppose we have a function $f$defined by $y = f(x)$, where $f$ is differentiable and invertible.

Suppose that $g$ is the inverse function of $f$. Then, the Inverse Function states that $f’(x) = \frac{1}{g’(f(x))}$.

Letting $y = f(x)$and $x = g(y)$, we can translate the Inverse Function theorem into Leibniz’s notation by writing:

$f’(x) = \frac{1}{g’(f(x))} \frac{d}{dx}f(x) = \frac{1}{g'(y)} \frac{dy}{dx} = \frac{1}{\frac{d}{dy}g(y)} \frac{dy}{dx} = \frac{1}{(\frac{dx}{dy})}$

In the above equations, we can see how Leibniz’s Notation behaves similarly to a fraction, although it must be emphasized that the derivative is not a fraction.

## Leibniz’s Notation vs. Other Notations

Leibniz’s Notation is one popular notation for differentiation, but there are several others that are also frequently used in calculus. Consider the list of derivative notations below to get an understanding of their relationship.

Note that $y’$and $f’(x)$are pronounced respectively as “y prime” and “f prime of x”.

Each notation has its own strengths and weaknesses in different contexts. Understanding their differences can help guide your decision on which derivative notation will work best in a given circumstance. To begin, note that Leibniz’s notation lets us easily express the derivative of a function without employing the use of another variable or function.

For example, we can express the derivative of $x^3$simply as $\frac{d}{dx}(x^3)$.

Another benefit of Leibniz’s notation is that its notation is very suggestive. As mentioned before, Leibniz’s notation often behaves like a fraction, although it’s one. Its appearance as a fraction suggests different ways that it can be manipulated, particularly with problems that concern the Chain Rule, the Inverse Function theorem, and integration by parts. As long as you have a solid understanding of why Leibniz’s notation is not a fraction, it’s usually okay to manipulate Leibniz’s notation as you would a fraction. That is if that helps you develop an intuition for different procedures in differential and integral calculus.

LaGrange’s notation is most popularly used for derivative problems in function notation. For example, we used LaGrange’s notation earlier to express the Chain Rule.

To give another example, if we are given $f(x) = 3x^2 - 2x$, we can easily write $f’(x) = 6x - 2$.

In problems that use function notation, LaGrange is often the preferred choice, because it’s easier to mix up functions with function values when using Leibniz’s notation.

Finally, Newton’s notation is most often used in physics, and it’s usually reserved for derivatives with respect to time, like velocity and acceleration. Newton’s notation expresses derivatives by placing a dot over the dependent variable.

## Examples of Leibniz’s Notation

Let’s work through some examples together.

### Example 1

Let $y = \sqrt{4x+2}$. Find $\frac{dy}{dx}$. We'll need to use the Chain Rule in Leibniz's notation.

Set $y = \sqrt{u}$and $u = 4x+2$. Then, $\frac{dy}{du} = \frac{1}{2}u^{\frac{-1}{2}}$, and $\frac{du}{dx} = 4$.

Using the Chain Rule, we have:

$\frac{dy}{dx} = \frac{dy}{du} \frac{du}{dx} \frac{dy}{dx} = \frac{1}{2}u^{\frac{-1}{2}}\cdot4 \frac{dy}{dx} = \frac{4}{2\sqrt{u}} \frac{dy}{dx} = \frac{4}{2\sqrt{4x+2}} \frac{dy}{dx} = \frac{2}{\sqrt{4x+2}}$

### Example 2

Let $y = (6x+1)^2$. Find $\frac{dy}{dx}$.

We'll use the Chain Rule in Leibniz's notation.

Set $y = u^2$and $u = 6x+1$. Then, $\frac{dy}{du} = 2u$, and $\frac{du}{dx} = 6$.

Using the Chain Rule, we have:

$\frac{dy}{dx} = \frac{dy}{du} \frac{du}{dx} \frac{dy}{dx} = 2u \cdot 6 \frac{dy}{dx} = 12(6x+1) \frac{dy}{dx} = 72x+12$

### Example 3

Let $y = e^{3x+5}$. Find $\frac{dy}{dx}$.

We'll use the Chain Rule in Leibniz's notation.

Set $y = e^u$and $u = 3x+5$. Then $\frac{dy}{du} = e^u$and $\frac{du}{dx} = 3$.

Using the Chain Rule, we have:

$\frac{dy}{dx} = \frac{dy}{du} \frac{du}{dx} \frac{dy}{dx} = e^u \cdot 3 \frac{dy}{dx} = 3e^{3x+5}$

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