Consider the following. w = xy2 + x2z + yz², x = t², y = 5t, z = 5 (a) Find dw/dt using the appropriate Chain Rule. dw dt (b) Find dw/dt by converting w to a function of t before differentiating. dw dt
Consider the following. w = xy2 + x2z + yz², x = t², y = 5t, z = 5 (a) Find dw/dt using the appropriate Chain Rule. dw dt (b) Find dw/dt by converting w to a function of t before differentiating. dw dt
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Topic: Using the Chain Rule in Calculus**
**Objective: Understand how to calculate the derivative \( \frac{dw}{dt} \) using different methods.**
**Problem Statement:**
Consider the following function:
\[ w = xy^2 + x^2z + yz^2 \]
with the given expressions for the variables:
\[ x = t^2, \quad y = 5t, \quad z = 5 \]
**Tasks:**
(a) **Using the Chain Rule:**
Find \( \frac{dw}{dt} \) using the appropriate Chain Rule.
\[ \frac{dw}{dt} = \boxed{\phantom{ } } \]
(b) **Direct Substitution:**
Find \( \frac{dw}{dt} \) by converting \( w \) to a function of \( t \) before differentiating.
\[ \frac{dw}{dt} = \boxed{\phantom{ } } \]
**Instructions:**
- **Chain Rule Approach:** Calculate the derivative with respect to \( t \) by applying the Chain Rule to differentiate \( w = xy^2 + x^2z + yz^2 \) with respect to each variable \( x, y, \) and \( z \).
- **Direct Substitution Approach:** Substitute \( x = t^2 \), \( y = 5t \), and \( z = 5 \) into \( w \) first and then differentiate the resulting expression directly with respect to \( t \).
Ensure accurate calculations in both methods to confirm the consistency of the results.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F124169c6-8d32-433b-ba67-139925e22b98%2Fcd5525aa-d685-4904-98f5-e8fa27e5c595%2Fjszmh9m_processed.png&w=3840&q=75)
Transcribed Image Text:**Topic: Using the Chain Rule in Calculus**
**Objective: Understand how to calculate the derivative \( \frac{dw}{dt} \) using different methods.**
**Problem Statement:**
Consider the following function:
\[ w = xy^2 + x^2z + yz^2 \]
with the given expressions for the variables:
\[ x = t^2, \quad y = 5t, \quad z = 5 \]
**Tasks:**
(a) **Using the Chain Rule:**
Find \( \frac{dw}{dt} \) using the appropriate Chain Rule.
\[ \frac{dw}{dt} = \boxed{\phantom{ } } \]
(b) **Direct Substitution:**
Find \( \frac{dw}{dt} \) by converting \( w \) to a function of \( t \) before differentiating.
\[ \frac{dw}{dt} = \boxed{\phantom{ } } \]
**Instructions:**
- **Chain Rule Approach:** Calculate the derivative with respect to \( t \) by applying the Chain Rule to differentiate \( w = xy^2 + x^2z + yz^2 \) with respect to each variable \( x, y, \) and \( z \).
- **Direct Substitution Approach:** Substitute \( x = t^2 \), \( y = 5t \), and \( z = 5 \) into \( w \) first and then differentiate the resulting expression directly with respect to \( t \).
Ensure accurate calculations in both methods to confirm the consistency of the results.
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