Associated Topics || Dr. Math Home || Search Dr. Math

### Using the Definition of Derivative

```
Date: 9/1/95 at 21:31:15
From: Luke White
Subject: Calculus Homework- HELP!

To whom it may concern:

I am a high school senior at St. Joseph's High School in
South Bend, Indiana, and I am currently enrolled in BC Calculus.
Below are a few problems that you may be able to help with. I would be
extremely grateful. Thank you!

1.  Using the definition of derivative, find f'(x), in simplest form,
for f(x)=x^(1/3). (I know through the "tricks" that the answer is
(1/3)*x^(-2/3) but I can't get to that using the definition of
derivative.)

2. If f(x)= [x], prove or disprove that f(x) has a limit at x=1

Thank you very much for your time and effort on this! I am exceedingly
grateful.

With much thanks,
Luke White
```

```
Date: 9/4/95 at 11:38:32
From: Doctor Ken
Subject: Re: Calculus Homework- HELP!

Hello!

>1.  Using the definition of derivative, find f'(x), in simplest form, for
>f(x)=x^(1/3).  (I know through the "tricks" that the answer is
(1/3)*x^(-2/3)
>but I can't get to that using the definition of derivative.)
>
Hint:  When you have this-

x^(1/3) - (x+h)^(1/3)
Lim   ---------------------
h->0           h

try using the difference of two cubes, and multiplying the top and
bottom by x^(2/3) + x^(1/3)(x+h)^(1/3) + (x+h)^(2/3).  It works in
such a neat way!

>2. If f(x)= [x], prove or disprove that f(x) has a limit at x=1

Well, there are several ways that something can fail to have a limit
as it approaches a point:

a) the point could be completely isolated, with no place to approach from.

b) the point could have approaches on both sides but they could be different.

c) the point might have really really messy approaches (ones that you
couldn't draw with a normal pencil from Earth) and not even be
continuous in a neighborhood of the point.

Basically, the point has to be on a graph where the left side and the
right side both zoom in on the limit, and it's the SAME LIMIT ON BOTH
SIDES.  Think about that in the context of this problem.  Is your
teacher looking for a formal (delta - epsilon style) proof, or something
a little less formal?

- Doctor Ken,  The Geometry Forum

```
Associated Topics:
High School Calculus

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search