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Interpretation of dy/dx as a Rate Measurer

Recall that by the derivative ds / dt, we mean the rate of change of distance s with respect to the time t. In a similar fashion, whenever one quantity y varies with another quantity x, satisfying some rule y = f(x), then dy / dx (or f′(x)) represents the rate of change of y with respect to x and (dy/dx)x = x0(or f′(x0)) represents the rate of change of y with respect to x at x = x0 .
 
Further, if two variables x and y are varying with respect to another variable t, i.e., if x = f(t) and y = g(t), then by Chain rule dy/dx = (dy / dt) / (dx / dt), if dx / dt ≠ 0. Thus, the rate of change of y with respect to x can be calculated using the rate of change of y and that of x both with respect to t.





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