Find the derivative of \( y \) with respect \[ y=\ln \left(5 t e^{4 t}\right) \] \( \frac{d y}{d t}=\square \)

Find the first order derivative with respect to \( t \) for \( y=\ln(5te^{4t}) \). Evaluate the derivative by following steps: - step0: Evaluate the derivative: \(y^{\prime}=\frac{d}{dt}\left(\ln{\left(5te^{4t}\right)}\right)\) - step1: Use differentiation rules: \(y^{\prime}=\frac{d}{dg}\left(\ln{\left(g\right)}\right)\times \frac{d}{dt}\left(5te^{4t}\right)\) - step2: Find the derivative: \(y^{\prime}=\frac{1}{g}\times \frac{d}{dt}\left(5te^{4t}\right)\) - step3: Calculate: \(y^{\prime}=\frac{1}{g}\times \left(5e^{4t}+20te^{4t}\right)\) - step4: Substitute back: \(y^{\prime}=\frac{1}{5te^{4t}}\times \left(5e^{4t}+20te^{4t}\right)\) - step5: Multiply the terms: \(y^{\prime}=\frac{1+4t}{t}\) The derivative of \( y \) with respect to \( t \) is \( \frac{1+4t}{t} \).