Questions: Find the derivative of the function. y = x(x^3 + 1) y' =

Solution Steps

To find the derivative of the function \( y = x(x^3 + 1) \), we can use the product rule. The product rule states that the derivative of a product of two functions \( u(x) \) and \( v(x) \) is given by \( u'(x)v(x) + u(x)v'(x) \). Here, let \( u(x) = x \) and \( v(x) = x^3 + 1 \). We will find the derivatives \( u'(x) \) and \( v'(x) \), and then apply the product rule.

We start with the function given by \[ y = x(x^3 + 1). \]

To find the derivative \( y' \), we apply the product rule. Let \[ u = x \quad \text{and} \quad v = x^3 + 1. \] Then, the derivatives are \[ u' = 1 \quad \text{and} \quad v' = 3x^2. \] According to the product rule, we have: \[ y' = u'v + uv' = (1)(x^3 + 1) + (x)(3x^2). \]

Now we simplify the expression: \[ y' = x^3 + 1 + 3x^3 = 4x^3 + 1. \]

Final Answer