Question 1203750
<pre>
One approach is to find g(x), the inverse of f(x):

f(x) = {{{ sqrt(x^2 + 8x) }}},   {{{ x>=0 }}}

To avoid confusion, I'll use y=f(x), solve for y, and after that, we can write g(x):

  f(x) = {{{  y = sqrt(x^2 + 8x) }}}
Square both sides
  {{{ y^2 = (x^2 + 8x)  }}}
Add 16 to both sides, this completes the square on RHS
  {{{ y^2 + 16 = x^2 + 8x + 16 }}}
Factor RHS
  {{{ y^2 + 16 = (x+4)^2 }}} 
Take square root of both sides
  {{{ sqrt(y^2+16) = x + 4 }}}
Subtract 4 from both sides, and re-write with x on LHS:
  {{{ x = (sqrt(y^2+16)) - 4 }}}


Now we can write  {{{ g(x) = -4 + sqrt(x^2+16) }}} , and g(x) also has domain {{{x>=0}}}


g'(x) = {{{ (1/2)(x^2+16)^(-1/2) =  1/(2*sqrt(x^2+16)) }}} 


Just plug in x=3 to g(x) and g'(x) to get the desired values ( I got g(3)=1, g'(3)=1/10)