1. ## difference quotient help

I have 2 graded problems that are due monday, september 22.
The directions are to simplify the difference quotients of the following functions:
#1:
f(x) = (√x)
and
#2:
f(x) = 1/x
The difference quotient is: m = (f(x+h) - f(x)) / h
All work/steps must be shown.

I attempted both of the problems but did not get very far. I put the functions into the difference quotient equation but I don't know how to simplify the expressions properly.

#1:
f(x) = (√x)
(√(x+h) - √(x)) / h

#2:
f(x) = 1/x
(((1/(x+h)) - (1/x)) / h

Please respond quickly; all responses are greatly appreciated.

2. for #1 ... rationalize the numerator and simplify to determine the limit.

for #2 ... get a common denominator for the two fractions 1/x and 1/(x+h), combine them, then simplify. might be easier if you view the difference quotient like this ...

$\frac{1}{h}\left(\frac{1}{x+h} - \frac{1}{x}\right)$

3. 1. You can't really "simplify" this expression but you can change it in order to calculate the limit as h tends to zero. We begin with:

$\frac{\sqrt{x+h} - \sqrt{x}}{h}$

Multiply top and bottom by $\sqrt{x+h} + \sqrt{x}$:

$\frac{(\sqrt{x+h} - \sqrt{x})(\sqrt{x+h} + \sqrt{x})}{h(\sqrt{x+h} + \sqrt{x})}$

Distributively multiply out the top:

$\frac{x + h - x}{h(\sqrt{x+h} + \sqrt{x})}$

$\frac{h}{h(\sqrt{x+h} + \sqrt{x})}$

Cancel the factors of h in the top and bottom:

$\frac{1}{\sqrt{x+h} + \sqrt{x}}$

4. Originally Posted by skeeter
for #2 ... get a common denominator for the two fractions 1/x and 1/(x+h), combine them, then simplify. might be easier if you view the difference quotient like this ...
$\frac{1}{h}\left(\frac{1}{x+h} - \frac{1}{x}\right)$
I did this and it all simplified down to just -h. Can you try it and verify that this is the correct answer?

5. no, it does not simplify to just -h.

here's something to get you going in the right direction ...

$\frac{1}{h} \left(\frac{1}{x+h} - \frac{1}{x}\right) = \frac{1}{h} \left(\frac{x - (x+h)}{x(x+h)}\right)$

6. Yeah I got that part right. After that I simplify it to:
$(\frac{-h}{x^2+xh}) (\frac{1}{h})$
then
$(\frac{-h^2}{h(x^2+xh})(\frac{x^2+xh}{h(x^2+xh)})$
then
$\frac{-h^2(x^2+xh)}{h(x^2+xh)}$
then
$\frac{-h^2}{h}$
then
-h

What am I doing wrong?

7. you don't need a common denominator to multiply two fractions ... the h's cancel.

$\frac{1}{h} \left( \frac{-h}{x(x+h)}\right) = \frac{-1}{x(x+h)}$