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Differential Operator — Definition, Formula & Examples

A differential operator is a symbol or rule that tells you to take a derivative of whatever function follows it. The most common notation is DD or ddx\frac{d}{dx}, which acts on a function to produce its derivative.

A differential operator is a linear mapping D:Cn(R)→Cn−1(R)D: C^n(\mathbb{R}) \to C^{n-1}(\mathbb{R}) defined by D[f]=f′D[f] = f'. More generally, a linear differential operator of order nn is an expression of the form L=anDn+an−1Dn−1+⋯+a1D+a0L = a_n D^n + a_{n-1} D^{n-1} + \cdots + a_1 D + a_0, where aia_i are coefficients and DkD^k denotes taking the kkth derivative.

Key Formula

Dn[f(x)]=dnfdxnD^n[f(x)] = \frac{d^n f}{dx^n}
Where:
  • DD = The differential operator, meaning 'take the derivative'
  • nn = The order of differentiation (number of times to differentiate)
  • f(x)f(x) = The function being differentiated

How It Works

You treat DD as an algebraic symbol that means "differentiate." When you write D[f(x)]D[f(x)], you compute f′(x)f'(x). Applying DD twice, written D2[f(x)]D^2[f(x)], gives f′′(x)f''(x). This notation lets you manipulate derivatives algebraically, which is especially powerful in solving linear differential equations. For example, the equation y′′+3y′+2y=0y'' + 3y' + 2y = 0 can be rewritten as (D2+3D+2)[y]=0(D^2 + 3D + 2)[y] = 0, and you can then factor the operator as (D+1)(D+2)[y]=0(D+1)(D+2)[y] = 0 to find solutions.

Worked Example

Problem: Use the differential operator L=D2−5D+6L = D^2 - 5D + 6 to evaluate L[e2x]L[e^{2x}].
Step 1: Compute D[e2x]D[e^{2x}], the first derivative.
D[e2x]=2e2xD[e^{2x}] = 2e^{2x}
Step 2: Compute D2[e2x]D^2[e^{2x}], the second derivative.
D2[e2x]=4e2xD^2[e^{2x}] = 4e^{2x}
Step 3: Substitute into the operator expression L=D2−5D+6L = D^2 - 5D + 6.
L[e2x]=4e2x−5(2e2x)+6e2x=4e2x−10e2x+6e2x=0L[e^{2x}] = 4e^{2x} - 5(2e^{2x}) + 6e^{2x} = 4e^{2x} - 10e^{2x} + 6e^{2x} = 0
Answer: L[e2x]=0L[e^{2x}] = 0, which means e2xe^{2x} is a solution to the differential equation (D2−5D+6)[y]=0(D^2 - 5D + 6)[y] = 0.

Why It Matters

Differential operators turn differential equations into algebraic problems. In a course on ordinary differential equations, factoring operators like (D2−5D+6)=(D−2)(D−3)(D^2 - 5D + 6) = (D-2)(D-3) lets you read off solutions directly, the same way factoring a polynomial reveals its roots. This operator approach also extends to partial differential equations in physics and engineering through the del operator.

Common Mistakes

Mistake: Treating DD as a variable you can cancel rather than an operator that must act on a function.
Correction: DD is not a number — it represents an operation. You can factor and compose operators algebraically, but expressions like D2[f]=(D[f])2D^2[f] = (D[f])^2 are wrong. D2[f]D^2[f] means differentiate ff twice, not square the first derivative.