General derivative functions

Hi, I'm starting derivatives and I am getting confused. Some problems seem easier solved by

f(x)-f(a)

---------

x - a

As x approaches a

and some by

f(a+h)-f(a)

-----------

h

as h approaches 0

Why is this and how do I choose the best option? Does it matter if it's a velocity problem and not a tangent problem? I thought they were the same. Does it matter if I have a point (c,f(c)) to go off of? I apologize if I'm not making much sense, but I have practiced both methods and still don't understand why they are different.

Re: General derivative functions

The first one gives you the gradient between two points - a linear type profile.

The later can be used to find the gradient at any point on a curve - nonlinear

Re: General derivative functions

The two definitions are quite similar. Let $\displaystyle x = a+h$ for some small h and you'll see what I mean.

Re: General derivative functions

Ok. I think I understand. One is a secant line and one is a limit. Right? Or are they both limits? Are these two completely interchangeable? Say I have the function $\displaystyle g(x)=x^2+3x+2$. I get the same answer when using either method: $\displaystyle 2a+3$. However it was easier for me to use the $\displaystyle h\rightarrow0$ method because it didn't require polynomial division like the other one did.

Thank you for all your help.

Re: General derivative functions

They should be the same, or roughly the same...I don't see why one definition would be significantly different from the other.