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Calculus

Leibniz’s Notation & dy/dx Meaning

11.20.2021 • 6 min read

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.

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In This Article

  1. Who Is Leibniz?

  2. What Is Leibniz’s Notation System?

  3. What Is dy/dx?

  4. Leibniz's Notation vs. Other Notations

  5. Examples of Leibniz’s Notation

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 ff at xx is given by: ddxf(x)=limΔx0ΔyΔx=limΔx0f(x+Δx)f(x)Δx\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 ff defined by y=f(x)y = f(x), Leibniz's notation expresses the derivative of ff at xx as:

dydx\frac{dy}{dx}

It might be tempting to think of dydx\frac{dy}{dx}as a fraction. In fact, Leibniz himself first conceptualized dydx\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 dydx\frac{dy}{dx}as a quotient has been reinterpreted to align with the modern limit-based definition of a derivative. Now, dydyand dxdxare 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 ddx\frac{d}{dx}as an operator defined by the standard limit-based definition of a derivative. Suppose we have a function ffdefined by y=f(x)y = f(x). When we apply the operator ddx\frac{d}{dx}to yy, we have the expression ddxy\frac{d}{dx}y, or dydx\frac{dy}{dx}. This expression represents the derivative of yywith respect to xx(note that yyis the function value of ffat xx). In this way, the operator ddx\frac{d}{dx}takes in one function, and outputs another! More specifically, the operator ddx\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 ffand gg, and suppose that ggis differentiable at xxand ffis differentiable at g(x)g(x).

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

With these conditions satisfied, the Chain Rule states:

h(x)=f(g(x))g(x)h’(x) = f’(g(x))g’(x)

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

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

DY DX chart on why both versions of Chain Rule mean the same thing

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

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

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

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

f(x)=1g(f(x))ddxf(x)=1g(y)dydx=1ddyg(y)dydx=1(dxdy)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.

derivative variations in dx/dy

Note that yy’and f(x)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 x3x^3simply as ddx(x3)\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)=3x22xf(x) = 3x^2 - 2x, we can easily write f(x)=6x2f’(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=4x+2y = \sqrt{4x+2}. Find dydx\frac{dy}{dx}. We'll need to use the Chain Rule in Leibniz's notation.

Set y=uy = \sqrt{u}and u=4x+2u = 4x+2. Then, dydu=12u12\frac{dy}{du} = \frac{1}{2}u^{\frac{-1}{2}}, and dudx=4\frac{du}{dx} = 4.

Using the Chain Rule, we have:

dydx=dydududxdydx=12u124dydx=42udydx=424x+2dydx=24x+2 \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)2y = (6x+1)^2. Find dydx\frac{dy}{dx}.

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

Set y=u2y = u^2and u=6x+1u = 6x+1. Then, dydu=2u\frac{dy}{du} = 2u, and dudx=6\frac{du}{dx} = 6.

Using the Chain Rule, we have:

dydx=dydududxdydx=2u6dydx=12(6x+1)dydx=72x+12 \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=e3x+5y = e^{3x+5}. Find dydx\frac{dy}{dx}.

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

Set y=euy = e^uand u=3x+5u = 3x+5. Then dydu=eu\frac{dy}{du} = e^uand dudx=3\frac{du}{dx} = 3.

Using the Chain Rule, we have:

dydx=dydududxdydx=eu3dydx=3e3x+5\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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