Question

Please see http://i1163.photobucket.com/albums/q554/startanewww/firstprinciple.jpg
I can't follow the steps. How does the 2nd step go to the 3rd one?

Answer 1

in step 2 they have expandes

\[(x + \Delta x)^3 - x^3 = x^3 + 3x^2 \Delta x + 3x \Delta x^2 + \Delta x^3 - x^3\]

then simplified as you have

\[\Delta x ( 3x^2 + 3x \Delta x + \Delta x^2)\]

this is then split

\[\Delta x(x^2 + 2x \Delta x + \Delta x^2) + x^2 + x \Delta x + x^2\]

this then simplifies to

perfect square then x as a common factor and then x^2

\[\Delta x ( x + \Delta x)^2 + x(x + \Delta x) + x^2\]

you'll find there are 3 x^2 and 3 Deltax x terms