What is the difference between representing different ways of differentiation

I'm short of time but in brief:

1. represents the gradient of a tangent line to a curve at a point P i.e the gradient of the curve at P.

2. represents the gradient of a secant line between two points P and Q on a curve. If we consider calculating the value of as we make Q closer and closer to P, then that values approaches more and more closely.

e.g. pick two points P,Q on the curve with then by noting that the gradient of the secant PQ is given by rise/run, we have:

(*)

We can make Q approach P as near as we like by making h smaller, and then we can make the gradient of PQ as close to as we like. This is the idea of finding the limit of the gradient of PQ as h approaches 0. Since the limit as h approaches 0 is , we say that the gradient of the curve at P is i.e.



Note that we can't just say: Oh, we'll just set h=0 to find the gradient then! - that goes wrong, because if we set h=0, then:

a) algebraically the division by h that we did in (*) is undefined, and
b) geometrically, our picture of a triangle with a secant line hypotenuse and run of h and rise of is wrong since if h=0, then P and Q are on top of each other i.e. our triangle on which we were basing our whole argument no longer exists.

3. In scientific experiments, we often measure a finite change in a quantity when we make a finite change to another quantity e.g. we may change the voltage across a device by 0.01 V, and we see a change in the current through it of 0.03 A. Then we can find, approximately, the gradient of the I-V curve at V=0.01 V by calculating .

That's why you'll often see this notation in physics or engineering texts. For example, in electronics, the quantity is referred to as the "small signal resistance" - it's a measurable (or calculable) approximation to the gradient of the V-I curve of a device.