[thaumasiotes]

If a function f passes through some point (a,b), then the tangent to f through that point is given by

    (y-b) = f'(a)·(x-a)

and that function is affine but usually not linear. (For the tangent curve to be a linear function, you would need a·f'(a) = b, so that the tangent goes through the point (0,0).)

It's not at all obvious to me that this means that the function d(f) = df/dx is linear. It is linear, but I don't see how the tangent curve demonstrates it.

[giomasce]

It is common in this context to say "linear" to also mean "affine", because after all affine functions are not much more complicated than linear functions.

[amelius]

The derivative is only the direction of the tangent line. The affine part does not come into play.

[chombier]

The sum operator is linear, so its derivative is itself: the derivative of the sum is the sum of derivatives. Same goes for multiplication by a scalar.

From this you obtain that the derivative of any linear combination is the linear combination of the derivatives: differentiation is linear.