Question

Differentiation. Find the derivative of the given function. Use quotient rule.

f(x)= (x^-1)/(x+x^-1)

Answer 1

$f(x)=\frac{x^{-1}}{x+x^{-1}}$

$f'(x)=\frac{(x^{-1})'(x+x^{-1})-(x^{-1})(x+x^{-1})'}{(x+x^{-1})^2}=$$\frac{-x^{-2}(x+x^{-1})-(x^{-1})(1-x^{-2})}{(x+x^{-1})^2}=$$\frac{-x^{-1}-x^{-3}-x^{-1}+x^{-3}}{(x+x^{-1})^2}=$$\frac{-2x^{-1}}{(x+x^{-1})^2}=$$\frac{-2x}{(x^2+1)^2}$