What is the difference between representing different ways of differentiationWatch this thread
But i still have no idea what the difference is between representing the change as
and when d is triangle. Sometimes formualas will use both and they seem to mean different things. Or the first part of the question will use on and the second will use another for some purpose I can't figure out. :/
Differentiation is the process of seeing what happens when the amount of change gets progressively smaller and smaller (I'm sure you've seen this sort of limiting process), and it is only in the limit as the amount of change tends to zero that we should use dy/dx.
Maths texts should be fairly hot on the difference, but some physical sciences can play a bit more freely with the terminology.
Differentiation measures the rate of chang or something similar to that. When you do dy/dx it means differentiating y in respect to x. Same with any other thing in that form, i.e. dy/dt, or dt/dx. Could you give an example of a formula using the triangle?
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.