Linear approximations anyone?

Question

Use the linear approximation (1+x)^k=1+kx to find an approximation for the function f(x)=1/sqrt(4+x) for the values of x near zero.

This is a toughy...first correct answer who shows their work gets the 10 pts!!!

Answer 1

multiply top, bottom by 1/2
=1/2*(1/sqrt(1+x/4))
=1/2*(1+x/4)^-1/2

by your approximation, (1+x)^k=1+kx
so (1/2)(1+x)^k=(1/2)(1+kx)
so with k = -1/2 and plugging x/4 for the x, we have
1/2*(1+x/4)^-1/2 = (1/2)(1+(-1/2)x/4)=1/2-x/16

I thought linear approximations were f(x+deltax) = f(x)+deltaxf'(x)...