Difference quotients were introduced in an earlier lesson:
Prerequisites: function review, difference quotients.
Evaluating difference quotients requires a high degree of comfort working with
functions and function notation,
and this earlier lesson offers a thorough review.
Then, this current lesson gives practice
with difference quotient problems involving reciprocal functions and square roots.
DIFFERENCE QUOTIENTS
Let $\,f\,$ be a function
and let $\,h\,$ be a nonzero number.
A difference quotient is an expression of the form $$\cssId{s9}{\frac{f(x+h)f(x)}h}\ ,$$ which is a simplified version of: $$\cssId{s11}{\frac{f(x+h)f(x)}{(x+h)x}}$$ This expression gives the slope of the line through the points: Since the expression is a quotient (division) of differences (subtractions), the name difference quotient is appropriate. 

$\displaystyle\frac{f(x+h)f(x)}h$  the desired difference quotient 
$\displaystyle = \frac{\frac 1{x+h}  \frac 1x}h$  function evaluation; use definition of $\,f\,$ 
$\displaystyle = \frac{\frac 1{x+h}\cdot\frac xx  \frac 1x\cdot\frac{x+h}{x+h}}h$  get a common denominator 
$\displaystyle = \frac{\frac x{x(x+h)}  \frac {x+h}{x(x+h)}}h$  simplify 
$\displaystyle = \frac{\frac {x  (x+h)}{x(x+h)}}h$  write the numerator as a single fraction 
$\displaystyle = {\frac {xxh}{x(x+h)}}\cdot\frac 1h$ 
distributive law; and, dividing by $\,h\,$ is the same as multiplying by $\,\frac 1h\,$ 
$\displaystyle = {\frac {h}{x(x+h)h}}$  $x  x = 0\,$; and multiply across 
$\displaystyle = {\frac {1}{x(x+h)}}$ 
cancel: since $\,h\ne 0\,$, $\,\frac hh = 1\,$ Now, the factor of $\,h\,$ is gone in the denominator! 
As $\,h\,$ approaches zero,
$\,\displaystyle {\frac {1}{x(x+h)}}\,$ approaches $\,\displaystyle\frac{1}{x(x+0)}\,$,
which equals $\,\displaystyle \frac 1{x^2}\,$. The formula $\displaystyle\,\frac{1}{x^2}\,$ gives the slopes of the tangent lines to the graph of $\displaystyle\,f(x) = \frac 1x\,$! For example, the slope of the tangent line to $\displaystyle\,f(x) = \frac 1x\,$ at the point $\displaystyle\,(3,\frac 13)\,$ is $\displaystyle\,\frac{1}{3^2} = \frac 19\,$. Click ‘Submit’ on the WolframAlpha widget below to explore the power of Wolfram Alpha! You can create your own widgets by clicking here! 
On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. 
PROBLEM TYPES:
