Question 365178
<pre>
Difference quotient = {{{("t(x+h)"-"t(x)")/h}}}

First calculate t(x+h)

t(x) = x³ + 2

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

x³ + 2hx² + h²x + hx² + 2h²x + h² + 2 =

x³ + 3hx² + 3h²x + h³ + 2

Now substitute {{{(x^3 + 3hx^2 + 3h^2x + h^3 + 2)}}} for {{{"t(x+h)"}}}

and {{{(x^3+2)}}} for {{{t(x)}}}

in the difference quotient expression:

{{{("t(x+h)"-"t(x)")/h}}}
   
{{{((x^3 + 3hx^2 + 3h^2x + h^3 + 2)-(x^3+2))/h}}}

Remove the parentheses on top:

{{{(x^3 + 3hx^2 + 3h^2x + h^3 + 2-x^3-2)/h}}}
  
Cancel the x³ with the -x³ and the +2 with the -2:

{{{(cross(x^3) + 3hx^2 + 3h^2x + h^3 + cross(2)-cross(x^3)-cross(2))/h}}}

{{{(3hx^2 + 3h^2x + h^3)/h}}}

Factor h out of the top:

{{{(h(3x^2 + 3hx + h^2))/h}}}

Cancel the h's

{{{(cross(h)(3x^2 + 3hx + h^2))/cross(h)}}}

{{{3x^2 + 3hx + h^2}}}

That's it!

Edwin</pre>