What is the definition of Difference Quotient and 
how do you arrive at an answer when you have 
f(x) = x²+3x and 
DQ = ((f(x+h)-f(x))/h

The difference quotient is defined as the slope of 
the line through the two points (x, f(x)) and 
(x+h, f(x+h))

          y2 - y1
DQ = m = ---------
          x2 - x1

      f(x+h) - f(x) 
DQ = --------------- 
       (x+h) - x

      f(x+h) - f(x) 
DQ = --------------- 
        x + h - x

      f(x+h) - f(x) 
DQ = --------------- 
            h

To find that for the given f(x):


      f(x+h) - f(x) 
DQ = --------------- 
            h

First we calculate the red part,

f(x) = x² + 3x

Replace all the x's by (x+h)

f(x+h) = (x+h)² + 3(x+h)
       = (x+h)(x+h) + 3(x+h)
       = x² + 2hx + h² + 3x + 3h 

Now the blue part is given as

f(x) = x² + 3x  

So in

      f(x+h) - f(x) 
DQ = --------------- 
            h

replace the red f(x+h) by (x²+2hx+h²+3x+3h) and 
the blue f(x) by (x²+3x)

      (x²+2hx+h²+3x+3h) - (x²+3x) 
DQ = ----------------------------- 
                 h

      (x²+2hx+h²+3x+3h) - (x²+3x) 
DQ = ----------------------------- 
                 h


      x² + 2hx + h² + 3x + 3h - x² - 3x 
DQ = ----------------------------------- 
                  h

Combine like terms:

      2hx + h² + 3h 
DQ = --------------- 
            h

Factor h out of the top:

      h(2x + h + 3) 
DQ = --------------- 
            h

Cancel the h's

      1
      h(2x + h + 3) 
DQ = --------------- 
            h
            1

DQ = 2x + h + 3

Edwin