calculus
posted by kale .
limit as delta x approaches 0 when f(x) = 4/ sqr x
Have to use the (F(x+delta x)f(x))/delta x method

just plug and chug
[f(x+h)f(x)]/h
4(x+h)^(1/2)  4*x^(1/2)
Now, to evaluate (x+h)^(1/2), use the binomial theorem:
(a+b)^n = a^n + n*a^(n1)*b^1 + n(n1)/2 * a^(n1)b^2 + ...
(x+h)^(1/2) = x^(1/2) + (1/2)*x^(3/2)*h + (1/2)(3/2)/2 * x^(5/2)h^2 + ...
(x+h)^(1/2)  x^(1/2) = (1/2)*x^(3/2)*h + (1/2)(3/2)/2 * x^(5/2)h^2 + ...
divide all that by h to get
(1/2)*X^(3/2) + (1/2)(3/2)/2 * x^(5/2)h + ...
= 1/2x^(3/2) + 3/8x^(5/2)*h + ...
As h>0, all the terms with h disappear, leaving just
1/2 x^(3/2)
multiply that by 4 from the original function, and you get
2x^3/2 = 2/√x^{3}