When to use power rule vs quotient rule in derivatives?

[chemistry]

I'm confused about what to look for when solving for the derivative of a function of an expression.
For the function h(x) = 1/3x³ you use the power rule, similarly with the expression y = -4/x⁹
But with the function f(x) = (x+1)/√ x the quotient rule is used instead.
I'm very confused what scenarios require the power rule vs which scenarios require the product rule.

 

[pka]

I'm confused about what to look for when solving for the derivative of a function of an expression.
For the function h(x) = 1/3x³ you use the power rule, similarly with the expression y = -4/x⁹
But with the function f(x) = (x+1)/√ x the quotient rule is used instead.
I'm very confused what scenarios require the power rule vs which scenarios require the product rule.

I would not use the quotient rule: \(\displaystyle \dfrac{x+1}{\sqrt{x}}=x^{\frac{1}{2}}+x^{\frac{-1}{2}}\).

On the other hand for \(\displaystyle \dfrac{x+1}{\sqrt{x} +1}\) we must use it.