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Not strictly true. You need to distinguish between Δy, a small increment in y, and dy/dx, the rate of change of y with respect to x. If y is a distance and x is time, Δy is still a distance (meters) but dy/dx is a speed (meters per second), so the meaning of dy * k is not obvious, whereas Δy * k may well be meaningful.Consider a staircase. It made from horizontal (Δx) and vertical (Δy) segments. Suppose for simplicity that they are equal, 10 cm lengths. Δy/Δx = 1, obviously. I want to rise 10 m. To make 100 steps you need 2 x 10 x 100 = 2000 cm of wood. But to make a 45-degree ramp (dy/dx = 1) of the same height, you only need 1414 cm of wood.
You should have been my maths / physics teacher: then I'd have actually understood all this the first time round and not had to re-teach myself later...
what if we divide it by dt e.g. so it becomes speed and it has a meaning, is that ok ? ([dy/dt] * k = something / dt).