Calculus 3 Partial Derivatives. Question 1: Read Example 5 (p. 827). Based on the question, how can you tell which partial derivative should be used for the solution? Explain how this connects with your answer from Partial Derivatives of a Function of Two Variables. Chapter 14 Partial DerivativesEXAMPLE 4 Find az/ax assuming that the equationyz - In z = x + ydefines z as a function of the two independent variables x and y and the partial derivativeexists.Solution We differentiate both sides of the equation with respect to x, holding y constantand treating z as a differentiable function of .x:aax(xz)-ax In z = +azy axy-1 azz axax dyaxax ax= 1 + 014.3 Partial Derivatives 827= 1yzWith y constant,ax(32) =818 Planex=1(1, 2, 5)SurfaceTangentlineFIGURE 14.19 The tangent to the curveof intersection of the plane x-1 andsurface z = x² + y² at the point (1, 2, 5)(Example 5).EXAMPLE 5 The plane x = 1 intersects the paraboloid z = x² + y² in a parabola.Find the slope of the tangent to the parabola at (1, 2, 5) (Figure 14.19).Solution The parabola lies in a plane parallel to the yz-plane, and the slope is the valueof the partial derivative az/ay at (1, 2):-(x²³ ² + y²)= 2y(1.2)(1.2)(1,2)As a check, we can treat the parabola as the graph of the single-variable functionz = (1)² + y² = 1 + y² in the plane x = 1 and ask for the slope at y = 2. The slope,calculated now as an ordinary derivative, isEXAMPLE 6Functions of More Than Two VariablesThe definitions of the partial derivatives of functions of more than two independent vari-ables are similar to the definitions for functions of two variables. They are ordinary deriva-tives with respect to one variable, taken while the other independent variables are heldconstant.thenafdz= 2(2) = 4.-$12-- - ²5 - 4d (1 + y²)(²)| 1-²2yy-2If x, y, and z are independent variables andf(x, y, z)=xsin (y + 32),ə[x sin (y + 3z)] = xsin (y + 3z)= x cos (y + 32) (y + 32)= 3x cos (y + 3z).x held constantChain ruley held constant

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Calculus 3 Partial Derivatives. Question 1: Read Example 5 (p. 827). Based on the question, how can you tell which partial derivative should be used for the solution? Explain how this connects with your answer from Partial Derivatives of a Function of Two Variables.

Calculus 3 Partial Derivatives. Question 1: Read Example 5 (p. 827). Based on the question, how can you tell which partial derivative should be used for the solution? Explain how this connects with your answer from Partial Derivatives of a Function of Two Variables.

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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