How to differentiate expression involving surd y=x√(x+1) using product rule

Given y = uv, where u = f(x) and v = g(x), then y' = u'v + uv'.

Some questions involving the product rule may also involve other rules, such as the chain rule.

In this example, we want the gradient function of y = x√(x+1).

Before differentiating, express the function using a fractional index to represent the surd.

y = x × (x+1)^(1/2). The function is a product of two simpler functions, hence the product rule is applied.

Let u = x and v = (x + 1)^(1/2). Then u' = 1 and v' = (1/2) × (x + 1)^(-1/2) × 1 [v' by the chain rule]

y' = u' × v + u × v' → y' = 1 × (x + 1)^(1/2) + x × (1/2) × (x + 1)^(-1/2)

After simplification, we have y' = (2x + 3) / (2√(x+1))