differentiate using quotient rule the function: y = (3x + 1) / ((x + 1)^1/2))

The quotient rule states that the derivative of two functions, say u and v, is given by: (v * du/dx - u * dv/dx) / (v^2).

Here, we have: u = 3x + 1 v = (x + 1)^(1/2)

First, we need to find the derivatives of u and v.

du/dx = d(3x + 1)/dx = 3 dv/dx = d((x + 1)^(1/2))/dx = 1/2 * (x + 1)^(-1/2)

Now, we can substitute these into the quotient rule formula:

dy/dx = ((x + 1)^(1/2) * 3 - (3x + 1) * 1/2 * (x + 1)^(-1/2)) / ((x + 1))

Simplify the above expression to get the final derivative.