Find dy/dx when y = x(4x + 1)^1/2

Here we can use the product rule where dy/dx = v du/dx + u dv/dx.We let u = x and v = (4x + 1)^1/2 which means we get du/dx = 1 and by using the chain rule we get dv/dx = 1/2(4x + 1)^-1/24 which simplifies to dv/dx = 2(4x + 1)^-1/2.Plugging these results into the equation for the product rule we get: dy/dx = (4x + 1)^1/2 + 2x(4x + 1)^-1/2.This result can also be simplified by taking out a factor of (4x + 1)^-1/2 to get dy/dx = (4x + 1)^-1/2((4x+1) +2x) which proves thatdy/dx = (4x + 1)^-1/2(6x +1). Remember that (4x + 1)^-1/2 can also be written as the square root on the denominator of a fraction.