Questions: Find the derivative. y = x^3 ln x

Solution Steps

To find the derivative of the function \( y = x^3 \ln x \), we apply the product rule. Let \( u = x^3 \) and \( v = \ln x \). The derivatives are \( u' = 3x^2 \) and \( v' = \frac{1}{x} \).

Using the product rule, we have: \[ \frac{dy}{dx} = u'v + uv' = (3x^2)(\ln x) + (x^3)\left(\frac{1}{x}\right) \] This simplifies to: \[ \frac{dy}{dx} = 3x^2 \ln x + x^2 \]

Final Answer