The equations represent the same thing, the only difference between them being the notation used.

The derivative of a function `f(x)` at a given point` x = c` , where `c in (a,b)` , represents the slope of the tangent line to the curve `f(x)` , at `x = c.` The slope of the tangent line is evaluated using the derivative of the function, such that:

`lim_(x->c) (f(x) - f(c))/(x - c) = f'(c)`

Since `x -> c` yields that `x - c -> 0` (solve as you solve an equation)

Replacing `Delta x` for `x - c,` and since` x - c -> 0` yields `Delta x -> 0,` such that:

`lim_(x->c) (f(x + Delta x) - f(x))/(Delta x) = f'(x)`

**Hence, comparing both equations,after performing the indicated
conversions, yields that they coincide.**

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