Higher order derivatives

Hello,

In my begining calculus class, my professor explained the process for finding higher order derivatives of a function, the derivative of a derivative. And eventually we reach a point were the function is derived to a constant value and since the derivative of a constant is zero, we cannot proceed any further. So my question is this: What is the advantage or purpose of finding the higher order derivatives of a function? I can only guess; does this allow us to be more precise in describing the curve of the function or the slope of the tangent line at x=a?

Polynomial functions have the property you describe of having higher order derivatives go to 0. Transcendental functions like exp(x), sin θ, and cos θ do not behave in that fashion. They have an infinite number of higher order derivatives.

Another example is when inspecting a function's local maximum's and minimum's.
The first derivative can tell you whether there's is one at one point.
The second derivative can tell you if it is a maximum or minimum.

Derivatives are also very useful when:

• solving differential equations
• calculating limits
• using polynomial interpolation
• and the list goes on...

Generally you won't need more than the third derivative in you everyday math, but for many advanced applications higher orders are needed.

When you learn about infinite series expansions for functions you'll understand from your knowledge of polynomials why some functions have an infinite number of non-zero derivatives.

One of the most useful things in the applied math of all science and engineering fields is Taylor's theorem and Taylor series, which require higher derivatives. They allow you to approximate (nearly) arbitrary functions with polynomials.

Thank you all for pointing me in the right direction! I'll pick it up and run with it.

Georacer said:

Another example is when inspecting a function's local maximum's and minimum's.
The first derivative can tell you whether there's is one at one point.
The second derivative can tell you if it is a maximum or minimum.
....

Click to expand...

Small correction: there may be a local minimum or maximum where the first derivative is zero. A horizontal line, for example, has a zero first derivative everywhere, but has no strict local minima or maxima. We can also have an inflection point--such as illustrated below--where the first two derivatives are zero.

Cary

I should be more careful with my statements, the level of the math forum seems to have taken a leap up.

I re-state:
The first derivative can tell you whether a point is a candidate for being a local minimum or maximum. In other words, it lets you find the critical points of the function. (I hope critical is the correct term. I translated directly from Greek).

As for the example, since the second derivative is zero too, aren't we hinted that the point isn't a minimum or maximum (what is the word that describes both?)?
In other words:
"The second derivative can tell you if it is a maximum or minimum or none"

Of course the above aren't valid for piecewise functions.

The point which is neither a minimum or a maximum is a "critical" point called a "point of inflection" or an "inflection point". The simplest example is the polynomial y = x^3. The first derivative is y = 3x^2 and the second derivative is y = 6x.

Setting the first derivative to 0 we see a solution at x = 0 with a multiplicity of 2, that is there are two zeros at the origin.

Setting the second derivative to 0 we see a single zero at the origin. For x > 0 we see that the original function is concave up, and for x < 0 the original function is concave down, thus identifying an inflection point.

ALL quadratic functions have 2nd derivatives that are constant and do not change sign, which implies that they cannot have "points of inflection".

@Georacer -- BTW I'm 100% sure that your English is better than my Greek

Καλύτερα από ελληνικά μου, επίσης.

Cary

I must say, I 'm touched that Greek characters appear in this forum.

P.S. Gotta love translate.google. It gives hilarious answers if you know the language sometimes.
In this case only an article is missing, but the phrase "There's money" translates in English to "No money".

I cannot tell a lie. Yes, I used Google translate. My knowledge of Greek is admittedly limited. When I went to university I had to learn the Greek alphabet forward and backward, and of course Greek symbols are used a lot in math. That's about it. You might say that the language is Greek to me. (Ouch, you say, having heard that far too many times.)

By the way, here's a tip for users of machine translation software. Use it to translate A->B, then B->A. If the second A makes sense, B's probably a reasonable translation; if not, rework the original A. Use short, simple sentences, commas to break up parts of sentences and present tense if possible. I need to translate between English and Spanish fairly often (and don't know Spanish either), and this approach seems to work pretty well.

Cary

It might sound funny to many of you, but we actually use the phrase "it sounds Chinese to me".

Also that would be a good time for a mod to move this whole conversation to the off-topic section.

Hello Pavlo if you are still with us there is one other comment about higher order derivatives worth making.

A = \(\frac{{{d^2}y}}{{d{x^2}}}\)

is not the same as

B = \({\left( {\frac{{dy}}{{dx}}} \right)^2}\)

A refers to the higher second derivative,

B refers to the first derivative squared.

You will see both as you progress. And before you ask, one use of the first derivative squared is in the formula for the length of a curve.

go well