Question 975410
<pre>
The other tutor did not use the definition, he just used the formula.

{{{matrix(3,2,
"","",
lim,   ("f(x+h)"-"f(x)")/h,    
"h->0", "")}}}{{{""=""}}}

{{{matrix(3,2,
"","",
lim,   (sqrt(1-8(x+h))-sqrt(1-8x))/h,    
"h->0", "")}}}{{{""=""}}} 

Multiply by {{{matrix(1,3,conjugate,of,numerator)/matrix(1,3,conjugate,of,numerator)}}}

{{{matrix(3,4,
"","","","",
lim,   ((sqrt(1-8(x+h))-sqrt(1-8x)))/h, ""*"" ,((sqrt(1-8(x+h))+sqrt(1-8x)))/((sqrt(1-8(x+h))+sqrt(1-8x))),     
"h->0", "","","")}}}{{{""=""}}} 

Multiply out numerator (FOIL):

{{{matrix(3,2,
"","",
lim,   ((1-8(x+h)^"")-(1-8x^""))/(h(sqrt(1-8(x+h)^"")+sqrt(1-8x))),    
"h->0", "")}}}{{{""=""}}} 

{{{matrix(3,2,
"","",
lim,   (1-8x-8h-1+8x)/(h(sqrt(1-8(x+h)^"")+sqrt(1-8x))),    
"h->0", "")}}}{{{""=""}}}

{{{matrix(3,2,
"","",
lim,   (-8h)/(h(sqrt(1-8(x+h)^"")+sqrt(1-8x))),    
"h->0", "")}}}{{{""=""}}}

Cancel the h's:

{{{matrix(3,2,
"","",
lim,   (-8cross(h))/(cross(h)(sqrt(1-8(x+h)^"")+sqrt(1-8x))),    
"h->0", "")}}}{{{""=""}}}



{{{matrix(3,2,
"","",
lim,   (-8)/(sqrt(1-8(x+h)^"")+sqrt(1-8x)),    
"h->0", "")}}}{{{""=""}}}

Now we can substitute 0 for h 

{{{(-8)/(sqrt(1-8(x+0)^"")+sqrt(1-8x))}}}{{{""=""}}}

{{{-8/(sqrt(1-8x)+sqrt(1-8x))}}}{{{""=""}}}

The bottom terms are like terms:

{{{-8/(2sqrt(1-8x))}}}

Cancel the 2 into the 8:

{{{-4/sqrt(1-8x)}}}

Edwin</pre>