When doing the difference quotient, the best way to
begin is by finding
 
           1
h(x+Δx) = ————
          x+Δx

Then find 

                  1      1
h(x+Δx) - h(x) = ———— - ———
                 x+Δx    x

Simplify that by getting an LCD:
                 
                 x-(x+Δx)      
h(x+Δx) - h(x) = ——————————
                  x(x+Δx)    

Simplify further:

                  x-x-Δx      
h(x+Δx) - h(x) = ————————
                  x(x+Δx)

Simplify further:

Next divide by Δx on the left by putting Δx 
under the left side.  Divide by Δx on the
right by multiplying by the reciprocal 1/(Δx): 
 

h(x+Δx) - h(x)        -Δx       1
——————————————  =  ———————— • —————— 
      Δx            x(x+Δx)     Δx

Cancel the Δx's on the right:

h(x+Δx) - h(x)        -Δx       1
——————————————  =  ———————— • —————— 
      Δx            x(x+Δx)     Δx

and you have:


h(x+Δx) - h(x)        -1       
——————————————  =  ————————  
      Δx            x(x+Δx)

And you can move the negative sign out in front:

h(x+Δx) - h(x)           1       
——————————————  =  - ————————  
      Δx              x(x+Δx)    

Edwin