Question

Ayuda!!! Con derivadas

Answer 1

1) 
$f(x)=x^{-1/2}\\ \\ f'(x)=-\dfrac{1}{2}x^{-1/2-1}\\ \\ \\ f'(x)=-\dfrac{1}{2}x^{-3/2}\\ \\ \\ f'(x)=-\dfrac{1}{2\sqrt{x^3}}\\ \\ \\ \boxed{f'(x)=-\dfrac{1}{2x\sqrt{x}}}$

2)
$f(x)=x^{-1}\\ \\ \\ f'(x)=-x^{-1-1}\\ \\ f'(x)=-x^{-2}\\ \\ \boxed{f'(x)=-\dfrac{1}{x^2}}$

3) 
$y=\dfrac{x^2-3}{x^2+3}\\ \\ \\ \dfrac{dy}{dx}=\dfrac{(x^2-3)'\cdot(x^2+3)-(x^2-3)\cdot(x^2+3)'}{(x^2+3)^2}\\ \\ \\ \dfrac{dy}{dx}=\dfrac{2x\cdot(x^2+3)-(x^2-3)\cdot2x}{(x^2+3)^2}\\ \\ \\ \dfrac{dy}{dx}=\dfrac{2x\cdot[(x^2+3)-(x^2-3)]}{(x^2+3)^2}\\ \\ \\ \boxed{\dfrac{dy}{dx}=\dfrac{12x}{(x^2+3)^2}}$