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Recipe

Test whether a function is an eigenfunction of the differentiation operator

Use when: you're handed a specific function and asked whether $D$ returns a scalar multiple of it

Goal: either a confirmed eigenvalue $ik$ , or a concrete reason the candidate function fails to be an eigenfunction of $D$

Taught in Differentiation as an operator, and the Fourier basis as its eigenfunctions

Quick recipe

Apply $D$ to the candidate function directly
Compare the result's shape to a constant multiple of the original
Read off the eigenvalue, once Step 2 confirms a match

1
Differentiate the given function using ordinary derivative rules, including the product or chain rule if it isn't a pure exponential.

Example

For $g(x)=xe^{ix}$ , the product rule gives $D[g](x)=e^{ix}+ix\,e^{ix}=(1+ix)e^{ix}$ .

Why: the definition of eigenfunction starts from what $D$ actually returns, not from a guess about the answer.
2
Check whether the differentiated expression equals one fixed constant times the exact original function, for every $x$ , with no extra terms gained or lost.

Example

$(1+ix)e^{ix}$ would need to equal $\lambda\cdot xe^{ix}$ for some constant $\lambda$ , but the lone $e^{ix}$ term has no factor of $x$ at all.
- the result equals (constant) $\times$ (original function), for every $x$ — it IS an eigenfunction; continue to Step 3.
- the result carries an extra term the original didn't have — it is NOT an eigenfunction; stop here.
Why: "eigenfunction" means returned unchanged in shape and only rescaled, so any structural mismatch rules it out immediately.
3
The constant multiplying the original function in Step 2 is the eigenvalue itself.

Example

For $e^{i3x}$ : $D[e^{i3x}]=3i\,e^{i3x}$ , so the eigenvalue is $3i$ , purely imaginary.

Why: this constant is exactly the $\lambda$ in $D[f]=\lambda f$ , the same role an eigenvalue plays for a matrix.

Common mistakes

Calling a function an eigenfunction just because differentiating it produces another familiar-looking trig function.
Missing a leftover factor from the product rule, like the $x$ in $xe^{ix}$ , and mistaking it for part of "the same function, rescaled."
Reporting a real eigenvalue for $D$ on this space, when every genuine eigenvalue of $D$ is purely imaginary.
Checking the match at one convenient value of $x$ instead of confirming it holds for every $x$ .