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Let c be a constant and let f and g be differentiable functions on R². Use the gradient rules V(cf)=cVf, V(f+g) = Vf+Vg, V(fg) = (f)g +fVg, and Choose the correct answer below. OA. xyz (5x+3y +2z +6yz,5x+3y + 2z+6xz,5x+3y +2z+6xy) OB. 6 xyz (5x+3y +2z) (5,3,2) C. 6xyz (5 +6yz(5x+3y + 2z),3+6xz(5x+3y + 2z),2 +6xy(5x+3y +2z)) OD. 6 xyz (5x+3y +2z) (yz,xz.xy) = gVf-fVg to find the gradient of f(x,y,z) = (5x+3y + 2z) e xyz.

Let c be a constant and let f and g be differentiable functions on R². Use the gradient rules V(cf)=cVf, V(f+g) = Vf+Vg, V(fg) = (f)g +fVg, and Choose the correct answer below. OA. xyz (5x+3y +2z +6yz,5x+3y + 2z+6xz,5x+3y +2z+6xy) OB. 6 xyz (5x+3y +2z) (5,3,2) C. 6xyz (5 +6yz(5x+3y + 2z),3+6xz(5x+3y + 2z),2 +6xy(5x+3y +2z)) OD. 6 xyz (5x+3y +2z) (yz,xz.xy) = gVf-fVg to find the gradient of f(x,y,z) = (5x+3y + 2z) e xyz.

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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**Understanding Gradients and Differentiable Functions** In this section, we explore the concept of differentiable functions and how to find their gradients. Consider the following scenario: Let \( c \) be a constant and let \( f \) and \( g \) be differentiable functions on \(\mathbb{R}^2\). We use the gradient rules: - \(\nabla (cf) = c \nabla f\) - \(\nabla (f + g) = \nabla f + \nabla g\) - \(\nabla (fg) = (\nabla f)g + f \nabla g\) - \(\nabla \left(\frac{f}{g}\right) = \frac{g \nabla f - f \nabla g}{g^2}\) We aim to find the gradient of the function \( f(x, y, z) = (5x + 3y + 2z) e^{6xyz} \). **Choose the correct answer:** - **A.** \( e^{6xyz} (5x + 3y + 2z + 6yz, 5x + 3y + 2z + 6xz, 5x + 3y + 2z + 6xy) \) - **B.** \( 6 \, e^{6xyz}(5x + 3y + 2z)(5,3,2) \) - **C.** \( e^{6xyz} (5 + 6yz(5x + 3y + 2z), 3 + 6xz(5x + 3y + 2z), 2 + 6xy(5x + 3y + 2z)) \) ✓ - **D.** \( 6 \, e^{6xyz}(5x + 3y + 2z) (yz, xz, xy) \) The correct answer is option **C**. Here, the gradient incorporates the partial derivatives calculated using both the original function and its exponential component.
Transcribed Image Text:**Understanding Gradients and Differentiable Functions** In this section, we explore the concept of differentiable functions and how to find their gradients. Consider the following scenario: Let \( c \) be a constant and let \( f \) and \( g \) be differentiable functions on \(\mathbb{R}^2\). We use the gradient rules: - \(\nabla (cf) = c \nabla f\) - \(\nabla (f + g) = \nabla f + \nabla g\) - \(\nabla (fg) = (\nabla f)g + f \nabla g\) - \(\nabla \left(\frac{f}{g}\right) = \frac{g \nabla f - f \nabla g}{g^2}\) We aim to find the gradient of the function \( f(x, y, z) = (5x + 3y + 2z) e^{6xyz} \). **Choose the correct answer:** - **A.** \( e^{6xyz} (5x + 3y + 2z + 6yz, 5x + 3y + 2z + 6xz, 5x + 3y + 2z + 6xy) \) - **B.** \( 6 \, e^{6xyz}(5x + 3y + 2z)(5,3,2) \) - **C.** \( e^{6xyz} (5 + 6yz(5x + 3y + 2z), 3 + 6xz(5x + 3y + 2z), 2 + 6xy(5x + 3y + 2z)) \) ✓ - **D.** \( 6 \, e^{6xyz}(5x + 3y + 2z) (yz, xz, xy) \) The correct answer is option **C**. Here, the gradient incorporates the partial derivatives calculated using both the original function and its exponential component.
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