Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
Related questions
Question
4.4 3
![**Objective:**
Find the derivative of the given function.
**Problem Statement:**
Given the function:
\[ y = -5e^{-4x^2} \]
**Task:**
Determine \(\frac{dy}{dx}\).
**Solution Explanation:**
To find the derivative of the function, apply the chain rule. The function is an exponential function of the form \( y = -5e^{u} \) where \( u = -4x^2 \).
**Steps:**
1. Differentiate \( e^u \) with respect to \( u \): \(\frac{d}{du}e^u = e^u\).
2. Differentiate \( u = -4x^2 \) with respect to \( x \):
\[ \frac{du}{dx} = \frac{d}{dx}(-4x^2) = -8x \]
3. Apply the chain rule:
\[ \frac{dy}{dx} = \frac{d}{du}(-5e^u) \cdot \frac{du}{dx} = -5e^u \cdot (-8x) \]
4. Substitute \( u = -4x^2 \):
\[ \frac{dy}{dx} = -5e^{-4x^2} \cdot (-8x) \]
\[ \frac{dy}{dx} = 40xe^{-4x^2} \]
Thus, the derivative \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = 40xe^{-4x^2} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F34d89729-f0b7-4660-b47f-8f1bb9a32e44%2Fc1655e6a-d860-43fb-ae44-6b6447821bd1%2Fb38ael_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Objective:**
Find the derivative of the given function.
**Problem Statement:**
Given the function:
\[ y = -5e^{-4x^2} \]
**Task:**
Determine \(\frac{dy}{dx}\).
**Solution Explanation:**
To find the derivative of the function, apply the chain rule. The function is an exponential function of the form \( y = -5e^{u} \) where \( u = -4x^2 \).
**Steps:**
1. Differentiate \( e^u \) with respect to \( u \): \(\frac{d}{du}e^u = e^u\).
2. Differentiate \( u = -4x^2 \) with respect to \( x \):
\[ \frac{du}{dx} = \frac{d}{dx}(-4x^2) = -8x \]
3. Apply the chain rule:
\[ \frac{dy}{dx} = \frac{d}{du}(-5e^u) \cdot \frac{du}{dx} = -5e^u \cdot (-8x) \]
4. Substitute \( u = -4x^2 \):
\[ \frac{dy}{dx} = -5e^{-4x^2} \cdot (-8x) \]
\[ \frac{dy}{dx} = 40xe^{-4x^2} \]
Thus, the derivative \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = 40xe^{-4x^2} \]
Expert Solution
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