Question
Find the derivative of \( y \) with respect \[ y=\ln \left(5 t e^{4 t}\right) \] \( \frac{d y}{d t}=\square \)
Ask by Perkins Coles. in the United States
Mar 09,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The derivative of \( y \) with respect to \( t \) is \( \frac{1+4t}{t} \).
Solution
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 \( \frac{1+4t}{t} \).
Answered by UpStudy AI and reviewed by a Professional Tutor
Like
error msg
Explain 
Simplify this solution Extra Insights
To find the derivative of \( y \) with respect to \( t \) where \( y = \ln(5t e^{4t}) \), we can use the properties of logarithms to simplify our work. First, we can break this down using the properties of logarithms: \[ y = \ln(5t) + \ln(e^{4t}) = \ln(5) + \ln(t) + 4t \] Now, we can differentiate term by term: \[ \frac{dy}{dt} = 0 + \frac{1}{t} + 4 = \frac{1}{t} + 4 \] Thus, the derivative of \( y \) with respect to \( t \) is: \[ \frac{dy}{dt} = \frac{1}{t} + 4 \]
