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This is a mathematical question not a beginners electronic question. when you differentiate y = mT + C the answer is m, the C drops out (not varying with T). For a step function, along the zero line the derivative is 0, then you get the rising edge which gives some function, as Y is varying with T. When you get to the flat top, again the derivative is 0 because before it was C (some value not linked to T).
Visualize a function that is similar to a unit step function, except that it goes linearly from 0 to 1 in a time equal to t. The derivative will be a pulse of amplitude 1/t and a duration of t. The area of the pulse will be (1/t) X t = 1. Now let the function go linearly from 0 to 1 in time t/2 (Half the time). The derivative is now a pulse with amplitude 1/(t/2) = 2/t and a duration of t/2. Its area is still 1, but its amplitude is 2. Continue decreasing the time period to zero. The amplitude of the pulse is now infinity and its duration is 0, but its area is still 1. This is the definition of an impulse function. This is not a rigorous proof, but simply an aid to visualizing what's going on.