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?
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.
I dont know how to edit a question, so I'll write it in the comments. The second equation is supposed to be:
f'(x)= the limit of [f(x)+delta x)-f(x)]/delta x as delta x approaches zero