Find the derivative of the following function by first expanding or simplifying the expression. \[ g(w)=\frac{3 e^{2 w}+4 e^{w}}{e^{w}} \] \( g^{\prime}(w)=\square \)

We are given the function \[ g(w)=\frac{3 e^{2w}+4e^{w}}{e^{w}} \] **Step 1: Simplify the function** Divide each term in the numerator by \( e^{w} \): \[ g(w)=\frac{3 e^{2w}}{e^{w}}+\frac{4 e^{w}}{e^{w}}=3e^{w}+4 \] **Step 2: Differentiate the simplified function** Differentiate \( g(w)=3e^{w}+4 \) with respect to \( w \): \[ g'(w)=\frac{d}{dw}\left(3e^{w}+4\right) \] Since \(\frac{d}{dw}(e^{w})=e^{w}\) and the derivative of a constant is 0: \[ g'(w)=3e^{w}+0=3e^{w} \] Thus, the derivative is \[ g'(w)=3e^{w} \]