I have just begun studying derivatives in calculus.  What I am incredibly confused about is the difference between the "Definition of the Tangelt Line with Slope m: The limit of delta y divided by delta x as delta x approaches zero = the limit of limit f(c + deltax)-f(c)/ delta x as delta x approaches zero= m. (If f is defined on an open interval containing c, and if the limit exists, then the line passing thtough (c,f(c)) with slope m is the tangelt line to the graph of f at the point (c,f(c)). and the definition of a derivative of a function, which is the same equation, but with all c's replaced with an x. (the derivative of f at x is given by this equation provided the limit exists. For all x for which this limit exists, f' is a function of x.  Δy  =  f(x+Δx) - f(x)   ___________  Δx Δx  What is the difference between the equation that uses c and the one that uses only x? What different things are these equations used to find?

Expert Answers

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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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