Where does the linearity of differential operators come from? I just quoted the linearity of a differential operator, namely d/dz, in a proof and I was wondering where the root of this linearity lies. All of the differential operators which I have encountered seem to be linear and the 'sketch' derivation in my Vector Calculus course for higher dimensional derivatives used a linear mapping approach. My question is, where does this linearity appear and how does it fit in with the natural 'rates of change' intuition of derivatives?

You can go back to the limit definition of the derivative.

$$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$

Let $k(x) = f(x) + g(x)$. Then,

$$k'(x) = \lim_{h \to 0} \frac{f(x+h) + g(x+h) - f(x) - g(x)}{h} = f'(x) + g'(x)$$

A similar argument shows that $[a f]'(x) = a f'(x)$ for a scalar $a$.