Question

Question #c75cc

Answer 1

`f'(x) = -(2a^2x)/(sqrt(a^2-x^2)(a^2+x^2)^(3/2))`

Explanation:

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))`