Tangents and the Derivative at a Point (Part II)

Tangents and the Derivative at a Point (Part II)

Acceleration is a vector quantity that describes the rate of change of velocity over time. In other words, it's how quickly an object is moving, and how quickly it's changing direction.

We are familiar with this notation, as well as commonly in the literature, where people use df(x), dx at the point x equals to zero. This means at the point x equals to x zero.

The d here, or in this notation, df, dx means the difference of the function at the point x equals to zero plus h and at the point x equals to x zero. Change in ix is dx.

Very commonly, this notation is written as dy/dx. So, suppose y equals f(x) is the function, if the derivative of the function at every point in its domain exists (i.e., if it has a derivative), then we can compute f'(x) at every point x.

Get quality help now

WriterBelle

Verified writer

Proficient in: Math

4.7 (657)

“ Really polite, and a great writer! Task done as described and better, responded to all my questions promptly too! ”

+84 relevant experts are online

Hire writer

This becomes a function in x, which we sometimes write as y' or dy/dx. The point to note is that f'(x) is actually a function depending on x while f'(x-nought) is a number.

Let's look at an example. The function f(x) = x2 can be differentiated to give the derivative f'(x) = 2x. If we look at f'(3), we find that it is equal to 6--a number.

Now, since f'(x) may be a function of x again, we can consider the second derivative of f.

Now, if we differentiate f'(x) again and take its second derivative with respect to x, we get what we call the second derivative or this notation.

Now this is of course very natural.You have two dashes here (differentiating two times),and the two here up in the numerator (also differentiating two times)and the two at the denominator here (differentiating two more times).

We are able to continue the process as far as we like, and in general the nth derivative of a function is denoted by this symbol. The nth derivative of a function is denoted by n. The n here means differentiate n times and the same meaning to this n here.

Why must we consider second and higher derivatives of functions? If we look at this example, the position of an object at time t is given by this function [s] equals to [s(t)].

The derivative of s with respect to t is the velocity of an object that is moving with respect to time. Thus, if we differentiate again this s'(t), that means we differentiate the velocity, which gives us the acceleration of the object.

The acceleration of an object is equal to the change in velocity divided by the change in time. The second derivative of s represents acceleration, and the third derivative of s represents jerk. The change of acceleration with respect to time.

Thus, we have the concept of differentiability and can consider whether a function is continuous at a certain point or not. Now, actually continuity implies differentiability, so we have this theorem that says that differentiability implies continuity.

Putting it another way, if the derivative of a function f at x equals 0, then the function must be continuous at this point x 0 as well.

The proof is as follows. We assume that the function has derivative f'(x nought) at the point x nought, which means that f'(x nought) exists. Now, by definition, what is f'(x nought)? It is equal to the limit of this ratio when h tends to zero: again, this limit exists.

The function f(x) plus x − 0 is not equal to the function f(x − 0). If we rewrite this equation into the following form that f(x) plus x − 0 equals f(x − 0) plus h times x, then consider h tends to zero. So, we take the limit when h tends to zero of the left-hand side which is equal to the limit of the right-hand side here.

We can divide the limit into three pieces. The first piece of the limit is simply equal to the number itself: no involving h. The second piece of the limit is exactly equal to f'(x nought), which we assume to exist.

The last bit here, when h tends to zero, of course, is equal to zero. So after all this we find that the expression on the left-hand side is equal to f(x). In other words, when h tends to zero, f(x) is continuous at x=0.