Notation for Differentiation: Leibniz, Lagrange, Newton

What Is Notation For Differentiation?

Notation for differentiation is the set of written forms used to express a derivative, the rate at which one quantity changes with respect to another. Four systems are in common use: Leibniz's $\frac{dy}{dx}$, Lagrange's $f'(x)$, Newton's $\dot{y}$ and Euler's $Df$. They all describe the same derivative and differ only in what they put on the page.

For $y = f(x) = x^3$, the same result reads four ways:

$$\frac{dy}{dx} = f'(x) = Df(x) = 3x^2$$

If the variable is time, $x = t^3$, Newton's form adds a fifth: $\dot{x} = 3t^2$. A student on Math Doctors once asked for "a general overview of the meanings of the different derivative notations", and this table is that overview.

A derivative only exists where its limit exists, whatever the notation. The full definition, and why the slope of a tangent is a limit, is in the guide to the derivative of a function.

How Does Leibniz Notation Work?

Leibniz notation writes the derivative of $y$ with respect to $x$ as $\frac{dy}{dx}$, which reads "d y by d x". It always names both the quantity being differentiated (on top) and the variable it changes with (underneath).

What do the d's mean? The $d$ stands for "difference", an infinitely small change. Leibniz pictured $dy$ and $dx$ as infinitesimal changes in $y$ and $x$; modern calculus defines the whole symbol as one limit instead:

$$\frac{dy}{dx} = \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x}$$

What Is The Difference Between $dy/dx$ And $d/dx$?

$\frac{d}{dx}$ is an operator: an instruction, "differentiate with respect to $x$", waiting for something to act on. $\frac{dy}{dx}$ is that operator already applied to $y$. So $\frac{d}{dx}(x^3) = 3x^2$ and, if $y = x^3$, $\frac{dy}{dx} = 3x^2$ say the same thing in two layouts.

How Are Higher Derivatives Written?

The second derivative is the derivative of the derivative, and the notation records exactly that:

$$\frac{d}{dx}\left(\frac{dy}{dx}\right) = \frac{d^2y}{dx^2}$$

The operator $d$ is applied twice, so the top gets $d^2$. The bottom $dx^2$ means $(dx)^2$, the variable used twice, not $d$ times $x^2$. That is the answer to the question readers keep sending in: "Why is the second derivative written as $d^2y/dx^2$?"

How Do You Evaluate A Leibniz Derivative At A Point?

Use a vertical evaluation bar with the point underneath:

$$\left.\frac{dy}{dx}\right\rvert_{x=1}$$

Writing $\frac{d}{dx}f(1)$ means something else entirely: $f(1)$ is a constant, so its derivative is $0$.

Leibniz notation earns its keep in the chain rule. If $y$ depends on $u$ and $u$ depends on $x$, then

$$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$$

which looks like fractions cancelling. It is not literal cancellation, but the layout reminds you which factor is still missing. The same bookkeeping makes implicit differentiation readable, because $\frac{dy}{dx}$ keeps visible which variable depends on which.

How Does Lagrange Notation Work?

Lagrange notation marks a derivative with a prime: $f'(x)$ reads "f prime of x", and $f'(a)$ is the derivative at the point $a$. It treats the derivative as a new function in its own right, built from $f$.

Higher orders add primes, then switch to a bracketed number once primes get hard to count:

$$f'(x),\quad f''(x),\quad f'''(x),\quad f^{(4)}(x),\quad f^{(n)}(x)$$

Some older books write $f^{\text{iv}}(x)$ in lower-case Roman numerals for the fourth derivative. The brackets matter: $f^{(2)}(x)$ is the second derivative, while $f^2(x)$ usually means $\big(f(x)\big)^2$.

Prime notation is the shortest to write, which is why it dominates rule summaries such as the power rule: $(x^n)' = nx^{n-1}$. Its weak spot is that it never says which variable the derivative is taken with respect to.

Does $f(x, y) = \frac{dy}{dx}$ or $f'(x)$? A Physics Forums thread asks exactly this. When a function has two inputs, $f'$ is ambiguous: a derivative with respect to $x$ or to $y$? In a differential equation such as $\frac{dy}{dx} = f(x, y)$, $f(x, y)$ is the slope formula, not a derivative of $f$. Use $\frac{dy}{dx}$ for the unknown's derivative and partial notation (below) for derivatives of $f$ itself.

How Does Newton's Dot Notation Work?

Newton's dot notation puts a dot over a quantity to mean its derivative with respect to time: $\dot{x} = \frac{dx}{dt}$, and two dots mean the second derivative, $\ddot{x} = \frac{d^2x}{dt^2}$. Newton called these rates fluxions, the speed at which a "flowing" quantity changes.

The dot is reserved for time. Writing $\dot{y}$ for $\frac{dy}{dx}$ when $x$ is not time breaks the convention and confuses any physics reader.

That restriction is also why physicists love it. Position $x(t)$, velocity $\dot{x}$ and acceleration $\ddot{x}$ fit on one line, and Newton's second law becomes $m\ddot{x} = F$, the same chain the guide to velocity and acceleration builds step by step. Dots also make curves given by parametric equations compact: the slope is $\frac{dy}{dx} = \frac{\dot{y}}{\dot{x}}$ wherever $\dot{x} \neq 0$.

How Does Euler's D Notation Work?

D notation writes differentiation as an operator letter: $Df$ is the derivative of $f$, $D^2 f$ the second derivative, and $D_x$ names the variable when that is unclear. It is usually called Euler notation, but it was introduced by Louis Arbogast, who did not invent derivatives; he invented the habit of treating "differentiate" as an object you can do algebra with.

Louis Arbogast (1759 to 1803, France) published "Calcul des derivations" in 1800. Its key move was to separate operator symbols from the functions they act on, so $D$ could be added, multiplied and factorised like a number.

That operator algebra is why D notation still runs linear differential equations. The equation $y'' - 3y' + 2y = 0$ becomes

$$(D^2 - 3D + 2),y = 0$$

$$(D - 1)(D - 2),y = 0$$

and the factors point straight to the solutions $e^{x}$ and $e^{2x}$, so $y = C_1 e^{x} + C_2 e^{2x}$. The inverse $D^{-1}$ stands for an antiderivative: $D^{-1}(6x^2) = 2x^3 + C$.

How Do You Write Partial Derivatives In Each Notation?

A partial derivative differentiates a function of several variables with respect to one of them while the others are held fixed. Each family has a partial version:

The curly $\partial$ was first used by Adrien-Marie Legendre in 1786 and brought into wide use by Carl Jacobi in 1841, according to the MacTutor record of calculus symbols. Watch the order in the mixed row: the subscript form reads left to right ($x$ first), while the Leibniz form reads right to left. For the smooth functions met in school and first-year courses the two mixed partials agree, so the order rarely changes the answer, but it does change what you wrote.

How Do You Convert Between Derivative Notations?

Converting between notations means rewriting the same quantity, never recomputing it. Use this table as a dictionary:

The chain-rule row is the strongest argument for knowing more than one system. In Leibniz form the rule nearly writes itself; in prime form you have to remember where each derivative is evaluated.

Which Notation For Differentiation Should You Use?

Use the notation your problem's variables already suggest, and switch when another one makes a step clearer. One Quora reader asks whether there is "a practical reason we still use Leibniz' confusing dy/dx notation", and the answer is yes: it is the only notation that shows both variables.

• Leibniz when variables depend on each other: the chain rule, implicit differentiation, related rates, substitution in integrals and separating variables in differential equations.

• Lagrange when you are working with named functions and the variable is obvious: rule summaries, curve sketching, Taylor polynomials.

• Newton only for derivatives with respect to time: mechanics, circuits, orbital motion.

• Euler (D) for operator methods: solving linear differential equations with constant coefficients.

A physics paper can use all four on one page. Reading each one fluently is the skill; picking a favourite is a matter of taste.

Examples Of Notation For Differentiation

Each example uses the notation the question gives, then converts or evaluates, moving from a one-line rewrite to a mixed partial.

Example 1

Write the derivative of $y = 5x^4$ in Leibniz, Lagrange and D notation.

The power rule gives the same result in every system:

$$\frac{dy}{dx} = 20x^3$$

$$y' = 20x^3$$

$$Dy = 20x^3$$

Final answer: $20x^3$ in every notation

Example 2

For $y = (3x^2 + 1)^5$, find $y'$ and evaluate it at $x = 1$.

In prime notation the tempting move is to apply the power rule to the outside and stop:

$$y' = 5(3x^2 + 1)^4$$

At $x = 1$ this gives $5 \cdot 4^4 = 1280$. Check the slope a different way: $3x^2 + 1$ is itself changing (at rate $6x$), so the outer power is being pushed along by a moving inside. A slope that ignores the inside's rate cannot be right.

Rewrite in Leibniz notation with $u = 3x^2 + 1$, so $y = u^5$:

$$\frac{dy}{du} = 5u^4$$

$$\frac{du}{dx} = 6x$$

$$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = 5u^4 \cdot 6x = 30x(3x^2 + 1)^4$$

$$\left.\frac{dy}{dx}\right\rvert_{x=1} = 30(1)(4)^4 = 7680$$

The Leibniz layout exposed the missing factor $\frac{du}{dx}$.

Final answer: $y' = 30x(3x^2 + 1)^4$, and $y'(1) = 7680$

Example 3

For $f(x) = x^3 - 6x^2$, find $f''(1)$ and write it in Leibniz notation.

$$f'(x) = 3x^2 - 12x$$

$$f''(x) = 6x - 12$$

$$f''(1) = 6 - 12 = -6$$

In Leibniz notation with $y = f(x)$ this is $\left.\frac{d^2y}{dx^2}\right\rvert_{x=1} = -6$. The negative value means the curve is concave down at $x = 1$, the idea the second derivative is used for.

Final answer: $f''(1) = -6$

Example 4

A particle's position is $x(t) = t^3 - 4t$ metres after $t$ seconds. Find $\dot{x}$ and $\ddot{x}$ at $t = 2$.

$$\dot{x} = 3t^2 - 4$$

$$\dot{x}(2) = 12 - 4 = 8$$

$$\ddot{x} = 6t$$

$$\ddot{x}(2) = 12$$

The velocity is $8$ metres per second and the acceleration $12$ metres per second squared. A common first instinct is to leave the dots off the final answer and write "$x = 8$", which turns a velocity back into a position.

Final answer: $\dot{x}(2) = 8$, $\ddot{x}(2) = 12$

Example 5

A curve is given by $x = t^2$, $y = t^3$. Use dot notation to find $\frac{dy}{dx}$, then evaluate it at $t = 2$.

$$\dot{x} = 2t, \qquad \dot{y} = 3t^2$$

$$\frac{dy}{dx} = \frac{\dot{y}}{\dot{x}} = \frac{3t^2}{2t} = \frac{3t}{2} \quad (t \neq 0)$$

$$\left.\frac{dy}{dx}\right\rvert_{t=2} = 3$$

Final answer: $\frac{dy}{dx} = \frac{3t}{2}$, which is $3$ at $t = 2$

Example 6

For $f(x, y) = x^2 y^3$, find $f_x$, $f_{xy}$ and $\frac{\partial^2 f}{\partial x,\partial y}$, and evaluate $f_x$ at $(1, 1)$.

$$f_x = \frac{\partial f}{\partial x} = 2xy^3$$

$$f_{xy} = \frac{\partial}{\partial y}\left(2xy^3\right) = 6xy^2$$

$$\frac{\partial^2 f}{\partial x,\partial y} = \frac{\partial}{\partial x}\left(3x^2y^2\right) = 6xy^2$$

$$f_x(1, 1) = 2$$

The two mixed partials match, as they do for smooth functions, even though they were reached in opposite orders.

Final answer: $f_x = 2xy^3$, $f_{xy} = \frac{\partial^2 f}{\partial x,\partial y} = 6xy^2$, $f_x(1, 1) = 2$

Why Does Differentiation Have Several Notations?

"Each notation solved a different problem" is the short answer. The derivative was invented twice, by Newton and Leibniz, and later reworked by mathematicians who wanted it to do new jobs, so each system carries the problem its author cared about.

• Leibniz wanted to track variables. His $\frac{dy}{dx}$ shows what changes with what, which made rules such as the chain rule feel natural and keeps it in general use today (the overview of notation for differentiation on Wikipedia traces all four systems).

• Newton wanted to describe motion. Fluxions measured how fast a quantity "flows" in time, so a dot over the moving quantity was enough.

• Lagrange wanted calculus without infinitesimals. In his 1797 "Theorie des fonctions analytiques" he aimed to found the subject "freed from all consideration of the infinitely small", treating $f'$ as a function derived from $f$ by algebra.

• Arbogast wanted to calculate with operators. Separating $D$ from the function let him factor and invert differentiation itself.

Knowing which problem each notation was built for tells you which one to reach for.

Where Is Each Notation Used In Real Life?

Every field that models change picks the notation that keeps its variables clear.

• Mechanics and aerospace: equations of motion are written with dots, such as $m\ddot{x} + kx = 0$ for a spring.

• Electrical engineering: circuit equations like $L\frac{di}{dt} + Ri = V$ are solved with D-operator methods.

• Economics: marginal cost is written $\frac{dC}{dq}$ or $C'(q)$, the rate of change of cost per extra unit.

• Machine learning: training uses partial derivatives $\frac{\partial L}{\partial w}$ of a loss $L$ with respect to each weight $w$.

Common Errors When Working With Notation For Differentiation

Most notation errors are reading errors: the maths is right, but the symbol says something different from what you meant.

Mistake 1: Differentiating After Substituting

Where it slips in: Finding the slope at a point, such as the derivative of $f(x) = x^2$ at $x = 3$.

Don't do this: Write $\frac{d}{dx}f(3) = \frac{d}{dx}(9) = 0$. Substituting first turns the function into a constant, and a constant has derivative $0$.

The correct way: Differentiate first, then substitute: $f'(3) = 6$, or $\left.\frac{dy}{dx}\right\rvert_{x=3} = 6$. The memorizer who knows "plug in the number" but not when hits this on the first evaluation question.

Mistake 2: Reading The Second Derivative As A Square

Where it slips in: Second-derivative questions written in Leibniz notation.

Don't do this: Treat $\frac{d^2y}{dx^2}$ as $\left(\frac{dy}{dx}\right)^2$. For $y = x^4$ that gives $16x^6$.

The correct way: Differentiate twice: $\frac{d^2y}{dx^2} = 12x^2$. The superscripts count how many times you differentiate. The second-guesser who got $12x^2$ often "fixes" it to $16x^6$ after seeing the squares, so trust the operator reading.

Mistake 3: Using Dots Or Primes Where The Variable Is Unclear

Where it slips in: Problems with more than one variable, or with time and space together.

Don't do this: Write $\dot{y}$ for $\frac{dy}{dx}$, or $f'$ for a function of both $x$ and $y$.

The correct way: Keep dots for time only and switch to $\frac{dy}{dx}$ or $\frac{\partial f}{\partial x}$ whenever the variable could be misread. Students meeting their first two-variable function usually keep writing primes out of habit, and the ambiguity only shows up when someone else reads the work.

Practice Problems On Notation For Differentiation

1. Write the derivative of $y = \sqrt{x}$ in Leibniz and Lagrange notation, then find $f''(4)$.

2. Rewrite $\frac{d^2s}{dt^2}$ in Newton's and Lagrange's notation.

3. For $x(t) = e^{-2t}$, find $\ddot{x}$.

4. For $g(x, y) = x^3 y^2 + y$, find $g_y$ and $g_{yx}$.

5. Use Leibniz notation and the chain rule to differentiate $y = \cos(x^2)$.

Answer to Question 1: $\frac{dy}{dx} = f'(x) = \frac{1}{2\sqrt{x}}$, and $f''(4) = -\frac{1}{32}$.

Answer to Question 2: $\ddot{s}$ in Newton's notation and $s''(t)$ in Lagrange's.

Answer to Question 3: $\ddot{x} = 4e^{-2t}$.

Answer to Question 4: $g_y = 2x^3 y + 1$ and $g_{yx} = 6x^2 y$.

Answer to Question 5: With $u = x^2$, $\frac{dy}{dx} = -\sin u \cdot 2x = -2x\sin(x^2)$.

Key Takeaways

• Notation for differentiation has four main systems: Leibniz $\frac{dy}{dx}$, Lagrange $f'(x)$, Newton $\dot{y}$ and Euler $Df$.

• All four name the same derivative, so converting between them is rewriting, never recomputing.

• Leibniz notation shows both variables, which is why it suits the chain rule and related rates.

• Newton's dots mean derivatives with respect to time only.

• Differentiate first and substitute second, and read $\frac{d^2y}{dx^2}$ as differentiating twice, not squaring.

Your Next Move

Work through practice problems 1 to 5; if problem 5 loses its $2x$ factor, return to Example 2, and if problem 4 comes out in the wrong order, reread the section on partial derivatives.

Want a live Bhanzu trainer to walk you through switching notations? Book a free demo class.

Read More

• Table of derivatives

• The limit definition of the derivative

• The product rule

• History of calculus