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We prove this from the definition of the derivative in terms of limits.

Let us say x increases by Δx amount and consequently, y increases by Δy, ie., y becomes (y+Δy). Thus derivative :

The converse is also true, about the changes in y and x. When y changes from y to (y+Δy) then due to the same relationship between x and y, x will change from x to (x+Δx). So derivative:

Hence the product:

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Alternately, the same can be proved using the y = f(x) and y+Δy = f(x+Δx) using the definition of derivatives by limits.

Let us say x increases by Δx amount and consequently, y increases by Δy, ie., y becomes (y+Δy). Thus derivative :

The converse is also true, about the changes in y and x. When y changes from y to (y+Δy) then due to the same relationship between x and y, x will change from x to (x+Δx). So derivative:

Hence the product:

=============================

Alternately, the same can be proved using the y = f(x) and y+Δy = f(x+Δx) using the definition of derivatives by limits.