We rewrite the expression as:
#f(x) = (sqrt(a^2 - x^2))/sqrt(a^2+x^2) = ((a^2-x^2)^(1/2))/(a^2+x^2)^(1/2)#
Using the quotient rule:
#f'(x) = ((1/2)(a^2-x^2)^(-1/2)(-2x)(a^2+x^2)^(1/2) - (a^2-x^2)^(1/2)(1/2)(a^2+x^2)^(-1/2)2x)/(a^2+x^2) =#
#= (-x(a^2-x^2)^(-1/2)(a^2+x^2)^(1/2)-x(a^2-x^2)^(1/2)(a^2+x^2)^(-1/2))/(a^2+x^2)=#
# = -x(a^2-x^2)^(-1/2)(a^2+x^2)^(-1/2)-x(a^2-x^2)^(1/2)(a^2+x^2)^(-3/2)=#
#= -x(a^2-x^2)^(-1/2)(a^2+x^2)^(-3/2)[(a^2+x^2)+(a^2-x^2)] =#
# = -2a^2x(a^2-x^2)^(-1/2)(a^2+x^2)^(-3/2)#
So, the final expression is:
#f'(x) = -(2a^2x)/(sqrt(a^2-x^2)(a^2+x^2)^(3/2))#