Find the derivative of the function. \[ h(x)=x^{2} \arctan (9 x) \] \( h^{\prime}(x)=\square \)

To find the derivative of the function \( h(x) = x^{2} \arctan(9x) \), we can use the product rule, which states that if you have two functions, say \( u(x) \) and \( v(x) \), the derivative is given by \( u'v + uv' \). Here, let \( u(x) = x^{2} \) and \( v(x) = \arctan(9x) \). 1. The derivative of \( u(x) \) is \( u'(x) = 2x \). 2. The derivative of \( v(x) \) using the chain rule is: \[ v'(x) = \frac{1}{1 + (9x)^{2}} \cdot 9 = \frac{9}{1 + 81x^{2}}. \] Now apply the product rule: \[ h'(x) = u'v + uv' = (2x) \arctan(9x) + (x^{2}) \frac{9}{1 + 81x^{2}}. \] Combine these to get: \[ h'(x) = 2x \arctan(9x) + \frac{9x^{2}}{1 + 81x^{2}}. \] So, the final answer is: \[ h'(x) = 2x \arctan(9x) + \frac{9x^{2}}{1 + 81x^{2}}. \]