Linear Algebra
10.135 min read

Derivative Notations

Several equivalent notations for the derivative appear in textbooks and papers. For a function ff evaluated at x0x_0: Lagrange notation f′(x0)f'(x_0), Leibniz notation dfdx∣x0\frac{df}{dx}\big|_{x_0} or dfdx(x0)\frac{df}{dx}(x_0), and subscript notation fx(x0)f_x(x_0) (common in multivariate contexts). All mean the same thing: the derivative at x0x_0.

The Leibniz notation dfdx\frac{df}{dx} is particularly useful because it keeps track of "derivative of ff with respect to xx," which matters in the chain rule and when there are multiple variables. You will see all of these notations — learn to recognize them all.

Formal View

Remark 10.2 — Derivative Notations
The derivative of ff at x0x_0 is written as: f′(x0)f'(x_0), dfdx(x0)\frac{df}{dx}(x_0), dfdx∣x=x0\frac{df}{dx}\big|_{x=x_0}, fx(x0)f_x(x_0), or Df(x0)Df(x_0). All are equivalent.

The notation dfdx\frac{df}{dx} (Leibniz) is most common in calculus and physics. The subscript notation fxf_x is common in partial derivatives (Chapter 11).

Why This Matters

Different fields use different notations — being fluent in all of them is essential for reading research papers.

  • Physics uses x˙\dot{x} for time derivatives and ∇\nabla for spatial gradients.
  • Statistics uses ∂ℓ∂θ\frac{\partial \ell}{\partial \theta} for score functions (derivative of log-likelihood).
  • The chain rule looks most natural in Leibniz notation: dydx=dydu⋅dudx\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}.

Quiz

Question 1

Which of the following is NOT a standard notation for the derivative of ff at x0x_0?

Question 2

f′(x0)f'(x_0) and dfdx∣x=x0\frac{df}{dx}\big|_{x=x_0} denote the same thing.

Common Mistakes

  • Confusing f′(x)f'(x) (derivative function) with f′(x0)f'(x_0) (derivative value at a specific point).
  • Misreading dfdx\frac{df}{dx} as a fraction — it is a limit, not a ratio of two separate quantities.