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Curly d

Curly d

The symbol ∂ used in the notation for partial derivatives.

 

 

See also

Del operator, multivariable calculus

Key Formula

∂f∂x\frac{\partial f}{\partial x}
Where:
  • ∂\partial = The curly d symbol, indicating a partial derivative
  • ff = A function of two or more variables
  • xx = The variable with respect to which you differentiate, while all other variables are held constant

Worked Example

Problem: Given f(x, y) = 3x²y + 5y³, find the partial derivative of f with respect to x using curly d notation.
Step 1: Write the partial derivative in curly d notation. The ∂ symbol tells you to treat y as a constant while differentiating with respect to x.
∂f∂x=∂∂x(3x2y+5y3)\frac{\partial f}{\partial x} = \frac{\partial}{\partial x}(3x^2 y + 5y^3)
Step 2: Differentiate 3x²y with respect to x, treating y as a constant.
∂∂x(3x2y)=6xy\frac{\partial}{\partial x}(3x^2 y) = 6xy
Step 3: Differentiate 5y³ with respect to x. Since it contains no x at all, it is a constant and its derivative is zero.
∂∂x(5y3)=0\frac{\partial}{\partial x}(5y^3) = 0
Step 4: Combine the results.
∂f∂x=6xy\frac{\partial f}{\partial x} = 6xy
Answer: The partial derivative is ∂f/∂x = 6xy. The curly d symbol ∂ signals that we held y constant throughout.

Another Example

Problem: For the same function f(x, y) = 3x²y + 5y³, find the partial derivative with respect to y.
Step 1: Write the partial derivative in curly d notation, now treating x as a constant.
∂f∂y=∂∂y(3x2y+5y3)\frac{\partial f}{\partial y} = \frac{\partial}{\partial y}(3x^2 y + 5y^3)
Step 2: Differentiate each term with respect to y, holding x constant.
∂∂y(3x2y)=3x2,∂∂y(5y3)=15y2\frac{\partial}{\partial y}(3x^2 y) = 3x^2, \quad \frac{\partial}{\partial y}(5y^3) = 15y^2
Step 3: Combine the results.
∂f∂y=3x2+15y2\frac{\partial f}{\partial y} = 3x^2 + 15y^2
Answer: The partial derivative is ∂f/∂y = 3x² + 15y².

Frequently Asked Questions

What is the difference between ∂ (curly d) and d (straight d)?
The straight d is used for ordinary derivatives of functions that depend on a single variable, like df/dx. The curly d (∂) is reserved for partial derivatives of functions that depend on two or more variables. Using ∂ signals that you hold all other variables constant while differentiating with respect to one chosen variable.
How do you type or pronounce ∂?
The symbol ∂ is typically pronounced as "partial" — so ∂f/∂x is read "the partial derivative of f with respect to x" or simply "partial f, partial x." On a computer, you can type it in LaTeX as \partial. Its Unicode code point is U+2202.

∂ (curly d) vs. d (straight d)

The curly d (∂) appears exclusively in partial derivatives, where a function depends on multiple variables and you differentiate with respect to one while holding the others constant. The straight d appears in ordinary derivatives, where the function depends on only one independent variable. If you see ∂f/∂x, you know f has at least two variables. If you see df/dx, the function f depends on x alone (or all other variables themselves depend on x, making it a total derivative).

Why It Matters

The curly d is essential notation throughout physics, engineering, and advanced mathematics whenever quantities depend on more than one variable. It appears in foundational equations like the heat equation, wave equation, and Maxwell's equations. Without a distinct symbol for partial differentiation, it would be ambiguous whether other variables are held constant or allowed to vary.

Common Mistakes

Mistake: Using the straight d when writing partial derivatives, writing df/dx instead of ∂f/∂x for a multivariable function.
Correction: Always use ∂ when the function depends on more than one independent variable and you are differentiating with respect to just one. The symbol choice communicates that other variables are held constant.
Mistake: Confusing ∂ with the lowercase Greek letter delta (δ).
Correction: Although they look similar, ∂ (curly d) denotes partial differentiation, while δ (delta) is a distinct Greek letter used for other purposes such as small variations or the Kronecker delta. They are different symbols with different meanings.

Related Terms

  • Partial Derivative — The operation that curly d notation represents
  • Del Operator — Vector operator built from partial derivatives using ∂
  • Multivariable Calculus — The branch of calculus where ∂ is used
  • Derivative — The single-variable counterpart using straight d
  • Chain Rule — Extended version uses ∂ for multivariable functions
  • Gradient — Vector of all partial derivatives of a function