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Find and Əx dy 8 for f(x, y) = 7(2x-4y+9)

Find and Əx dy 8 for f(x, y) = 7(2x-4y+9)

Calculus: Early Transcendentals
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...
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**Problem Statement** Find \(\frac{\partial f}{\partial x}\) and \(\frac{\partial f}{\partial y}\) for \(f(x, y) = 7(2x - 4y + 9)^8\). **Explanation** This problem requires finding the partial derivatives of the function \(f(x, y)\). - \(\frac{\partial f}{\partial x}\) represents the rate of change of the function \(f\) with respect to the variable \(x\). - \(\frac{\partial f}{\partial y}\) represents the rate of change of the function \(f\) with respect to the variable \(y\). The function given is \(f(x, y) = 7(2x - 4y + 9)^8\). You will apply the chain rule and the power rule to differentiate with respect to each variable. **Steps for Calculating Partial Derivatives** 1. **Differentiation with respect to \(x\):** - Differentiate the inner function \(u = 2x - 4y + 9\) with respect to \(x\) to get \(\frac{\partial u}{\partial x} = 2\). - Use the chain rule: \(\frac{\partial f}{\partial x} = 7 \cdot 8(2x - 4y + 9)^7 \cdot 2\). 2. **Differentiation with respect to \(y\):** - Differentiate the inner function \(u = 2x - 4y + 9\) with respect to \(y\) to get \(\frac{\partial u}{\partial y} = -4\). - Use the chain rule: \(\frac{\partial f}{\partial y} = 7 \cdot 8(2x - 4y + 9)^7 \cdot (-4)\). These steps outline how to find each partial derivative for the given function.
Transcribed Image Text:**Problem Statement** Find \(\frac{\partial f}{\partial x}\) and \(\frac{\partial f}{\partial y}\) for \(f(x, y) = 7(2x - 4y + 9)^8\). **Explanation** This problem requires finding the partial derivatives of the function \(f(x, y)\). - \(\frac{\partial f}{\partial x}\) represents the rate of change of the function \(f\) with respect to the variable \(x\). - \(\frac{\partial f}{\partial y}\) represents the rate of change of the function \(f\) with respect to the variable \(y\). The function given is \(f(x, y) = 7(2x - 4y + 9)^8\). You will apply the chain rule and the power rule to differentiate with respect to each variable. **Steps for Calculating Partial Derivatives** 1. **Differentiation with respect to \(x\):** - Differentiate the inner function \(u = 2x - 4y + 9\) with respect to \(x\) to get \(\frac{\partial u}{\partial x} = 2\). - Use the chain rule: \(\frac{\partial f}{\partial x} = 7 \cdot 8(2x - 4y + 9)^7 \cdot 2\). 2. **Differentiation with respect to \(y\):** - Differentiate the inner function \(u = 2x - 4y + 9\) with respect to \(y\) to get \(\frac{\partial u}{\partial y} = -4\). - Use the chain rule: \(\frac{\partial f}{\partial y} = 7 \cdot 8(2x - 4y + 9)^7 \cdot (-4)\). These steps outline how to find each partial derivative for the given function.
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