Expert: Paul Klarreich - 1/22/2007

Question
Hi Paul,

I'm studying Grade 12 Calculus.
Could you briefly give me an overview on derivatives? I am just starting to learn them and i know a derivative is the slope at a particular point of a function. i have learned to find the equation of the tangent as well.

But, why is it possible to find the derivative for all real numbers in the domain of a polynomial function?

Is it because polynomials are continuous and have no "holes", "vertical asymptotes" or restrictions?

and why can any polynomial function be differntiated term by term?

Thanks for your help,

Samantha

Answer
Questioner:  Samantha
Hi Paul,

I'm studying Grade 12 Calculus.
Could you briefly give me an overview on derivatives? I am just starting to learn them and i know a derivative is the slope at a particular point of a function. i have learned to find the equation of the tangent as well.

But, why is it possible to find the derivative for all real numbers in the domain of a polynomial function?

Is it because polynomials are continuous and have no "holes", "vertical asymptotes" or restrictions?

and why can any polynomial function be differntiated term by term?

Thanks for your help,

Samantha
.............................................
Hi, Samantha,

You are really asking me to teach you the whole course and there is a limit to how much I can type here.

But you will learn a few rules for derivatives AFTER you learn the definition and learn how to apply it.  [Small tip: Be VERY careful about parenthesizing.  And I mean really careful.]

Then you will apply those rules to functions and find that since you can:
A. Differentiate a power of x.
B. Differentiate a product of a constant and another differentiable function.
C. Differentiate a sum of differentiable functions term-by-term.

then you can differentiate a polynomial everywhere.

When the fun starts (I.e. applications, such as related rates and optimization problems) write me again.