Find \( g^{\prime}(x) \) if \( g(x)=5 x e^{3 x} \) \( g^{\prime}(x)=\square \)

To find \( g^{\prime}(x) \) for the function \( g(x) = 5x e^{3x} \), we will use the product rule, which states that if you have two functions \( u(x) \) and \( v(x) \), then \( (uv)' = u'v + uv' \). Here, let \( u(x) = 5x \) and \( v(x) = e^{3x} \). First, compute the derivatives: - \( u'(x) = 5 \) - \( v'(x) = 3e^{3x} \) (using the chain rule) Now apply the product rule: \[ g^{\prime}(x) = u'v + uv' = 5 \cdot e^{3x} + 5x \cdot 3e^{3x} \] This simplifies to: \[ g^{\prime}(x) = 5 e^{3x} + 15x e^{3x} \] Finally, factor out \( e^{3x} \): \[ g^{\prime}(x) = e^{3x}(5 + 15x) \] So, \( g^{\prime}(x) = e^{3x}(15x + 5) \).