Question

Can someone explain to me the difference and application of the two formulas

lim (f(x+h) - f(x))/h
h->0

lim (f(x) - f(a))/(x-a)
x->a

Also what does this relate to:
y = f'(a) (x-a) + f(a)

Answer 1

To make it clear I understand that the first equation is the definition of the derivative, where does the second formula come from?

Answer 2

they are identical, they just look different. take the second one, replace
\[x-a\] by
\[h\] and you will see that there is no difference

Answer 3

\[h\] and \[x-a\] are both different ways of expressing the change in x for the secant line of a function. The first equation shows the limit of the function as the change of x tends directly to zero. The second function shows the limit of the function as x tends to a; as x gets really close to a, the change in x also approaches zero. They are the same function, one just uses a new variable (h) to describe the change in x.