**Problem Statement:**Find the first and second derivatives of \( y = 14e^{-4x} \).**Solution:****Step 1: First Derivative**Given the function \( y = 14e^{-4x} \), we need to find the first derivative \(\frac{dy}{dx}\).The derivative of \( e^{u(x)} \) is \( e^{u(x)} \cdot \frac{du}{dx} \), where \( u(x) = -4x \).1. Calculate \(\frac{d}{dx}(-4x) = -4\).2. Use the chain rule:\[ \frac{dy}{dx} = 14e^{-4x} \cdot (-4) \]\[ \frac{dy}{dx} = -56e^{-4x} \]**Step 2: Second Derivative**Next, we find the second derivative \(\frac{d^2y}{dx^2}\).1. Start with the first derivative: \(\frac{dy}{dx} = -56e^{-4x} \)2. Again, use the chain rule:\[ \frac{d}{dx}(-56e^{-4x}) = -56 \cdot (e^{-4x} \cdot (-4)) \]\[ \frac{d^2y}{dx^2} = -56 \cdot -4e^{-4x} \]\[ \frac{d^2y}{dx^2} = 224e^{-4x} \]**Conclusion:**The first derivative of \( y = 14e^{-4x} \) is:\[ \frac{dy}{dx} = -56e^{-4x} \]The second derivative of \( y = 14e^{-4x} \) is:\[ \frac{d^2y}{dx^2} = 224e^{-4x} \]

Name: **Problem Statement:**Find the first and second derivatives of \( y =
Uploaded: 2024-03-20T06:47:22.000Z
Channel: Bartleby
Description: **Problem Statement:**Find the first and second derivatives of \( y = 14e^{-4x} \).**Solution:****Step 1: First Derivative**Given the function \( y = 14e^{-4x} \), we need to find the first derivative \(\frac{dy}{dx}\).The derivative of \( e^{u(x)}