Here's one of my own. Maybe I should publish it? A confused calculus student attempted to evaluate $$\frac{d}{dx} \left( 1^n + 2^n + 3^n + \cdots + (x-1)^n \right)$$ and got $$n1^{n-1} + n2^{n-1} + n3^{n-1} + \cdots + n(x-1)^{n-1} + \text{constant}.$$ That is in fact correct. The "constant" is a Bernoulli number.