Question 287735
Construct and Simplify the difference quotient {{{ (f(x+h)-f(x))/h }}}
for the function given by: {{{ f(x)=2x^2-x }}}

I am really having trouble with this type of problem.
<pre><font size = 4 color = "indigo"><b>
Let's do it piece by piece. 

{{{ (highlight(f(x+h))-f(x))/h }}}

Let's first find {{{highlight("f(x+h)"))}}} using

{{{ f(red(x))=2red(x)^2-red(x) }}}

In place of those red {{{red(x)}}}'s, let's put {{{red((x+h))}}}

{{{ f(red(x+h))=2red((x+h))^2-red((x+h)) }}}

Simplify the right side:

{{{ f(x+h)=2(x+h)^2-(x+h) }}}
{{{ f(x+h)=2(x+h)(x+h)-(x+h) }}}
{{{ f(x+h)=2(x^2+2hx+h^2)-x-h }}}
{{{ f(x+h)=2x^2+4hx+2h^2-x-h }}}
{{{ f(x+h)=2x^2+4hx+2h^2-x-h }}}

Replace the {{{f(x+h)}}} in

{{{ (highlight(f(x+h))-f(x))/h }}}

by {{{highlight(2x^2+4hx+2h^2-x-h) }}}

{{{ (highlight(2x^2+4hx+2h^2-x-h)-f(x))/h }}}

Next we need to replace replace the {{{highlight(f(x))}}} in 

{{{ (2x^2+4hx+2h^2-x-h-highlight(f(x)) )/h }}}

by what we are given for {{{"f(x)"}}}, namely {{{f(x)=highlight((2x^2-x) ) }}}

{{{ (2x^2+4hx+2h^2-x-h-highlight((2x^2-x)) )/h }}}

Now we simplify:

{{{ (2x^2+4hx+2h^2-x-h-(2x^2-x) )/h }}}

{{{ (2x^2+4hx+2h^2-x-h-2x^2+x )/h }}}

Cancel the {{{2x^2}}} and the {{{-2x^2}}}

{{{ (cross(2x^2)+4hx+2h^2-x-h-cross(2x^2)+x )/h }}}

{{{ (4hx+2h^2-x-h+x )/h }}}

Cancel the {{{-x}}} and the {{{""+x}}}

{{{ (4hx+2h^2-cross(x)-h+cross(x) )/h }}}

{{{ (4hx+2h^2-h )/h }}}

Factor {{{h}}} out of the numerator:

{{{ (h(4x+2h-1) )/h }}}

Cancel the {{{h}}}'s

{{{ (cross(h)(4x+2h-1) )/cross(h) }}}

{{{4x+2h-1}}}

That's it!

Edwin</pre>