Derivatives of the given function. f ( x , y , z ) = x 2 y 3 + z 4 x = s 2 y = s t 2 z = s 2 t (a) We have to calculate the primary derivatives, ∂ f ∂ x , ∂ f ∂ y , ∂ f ∂ z (b) We have to calculate, ∂ x ∂ s , ∂ y ∂ s , ∂ z ∂ s (c) We have to complete the derivatives ( ∂ f ∂ s ) by chain rule. ∂ f ∂ s = ∂ f ∂ x . ∂ f ∂ x + ∂ f ∂ y . ∂ y ∂ s + ∂ f ∂ z . ∂ z ∂ s Answer should be in terms of independent variable s, t.

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Answer 1

Question

Chapter 14.6, Problem 1E

To determine

Derivatives of the given function.

$\begin{array}{l} f (x, y, z) = x^{2} y^{3} + z^{4} \\ x = s^{2} \\ y = s t^{2} \\ z = s^{2} t \end{array}$

(a) We have to calculate the primary derivatives,

$\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}$

(b) We have to calculate,

$\frac{\partial x}{\partial s}, \frac{\partial y}{\partial s}, \frac{\partial z}{\partial s}$

(c) We have to complete the derivatives $(\frac{\partial f}{\partial s})$ by chain rule.

$\frac{\partial f}{\partial s} = \frac{\partial f}{\partial x} . \frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} . \frac{\partial y}{\partial s} + \frac{\partial f}{\partial z} . \frac{\partial z}{\partial s}$

Answer should be in terms of independent variable s, t.

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Answer 2

Question

Chapter 14.6, Problem 1E

To determine

Derivatives of the given function.

$\begin{array}{l} f (x, y, z) = x^{2} y^{3} + z^{4} \\ x = s^{2} \\ y = s t^{2} \\ z = s^{2} t \end{array}$

(a) We have to calculate the primary derivatives,

$\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}$

(b) We have to calculate,

$\frac{\partial x}{\partial s}, \frac{\partial y}{\partial s}, \frac{\partial z}{\partial s}$

(c) We have to complete the derivatives $(\frac{\partial f}{\partial s})$ by chain rule.

$\frac{\partial f}{\partial s} = \frac{\partial f}{\partial x} . \frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} . \frac{\partial y}{\partial s} + \frac{\partial f}{\partial z} . \frac{\partial z}{\partial s}$

Answer should be in terms of independent variable s, t.

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Answer 3

Question

Chapter 14.6, Problem 1E

To determine

Derivatives of the given function.

$\begin{array}{l} f (x, y, z) = x^{2} y^{3} + z^{4} \\ x = s^{2} \\ y = s t^{2} \\ z = s^{2} t \end{array}$

(a) We have to calculate the primary derivatives,

$\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}$

(b) We have to calculate,

$\frac{\partial x}{\partial s}, \frac{\partial y}{\partial s}, \frac{\partial z}{\partial s}$

(c) We have to complete the derivatives $(\frac{\partial f}{\partial s})$ by chain rule.

$\frac{\partial f}{\partial s} = \frac{\partial f}{\partial x} . \frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} . \frac{\partial y}{\partial s} + \frac{\partial f}{\partial z} . \frac{\partial z}{\partial s}$

Answer should be in terms of independent variable s, t.

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Answer 4

Question

Chapter 14.6, Problem 1E

To determine

Derivatives of the given function.

$\begin{array}{l} f (x, y, z) = x^{2} y^{3} + z^{4} \\ x = s^{2} \\ y = s t^{2} \\ z = s^{2} t \end{array}$

(a) We have to calculate the primary derivatives,

$\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}$

(b) We have to calculate,

$\frac{\partial x}{\partial s}, \frac{\partial y}{\partial s}, \frac{\partial z}{\partial s}$

(c) We have to complete the derivatives $(\frac{\partial f}{\partial s})$ by chain rule.

$\frac{\partial f}{\partial s} = \frac{\partial f}{\partial x} . \frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} . \frac{\partial y}{\partial s} + \frac{\partial f}{\partial z} . \frac{\partial z}{\partial s}$

Answer should be in terms of independent variable s, t.

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Answer 5

Chapter 14.6, Problem 1E

To determine

Derivatives of the given function.

$\begin{array}{l} f (x, y, z) = x^{2} y^{3} + z^{4} \\ x = s^{2} \\ y = s t^{2} \\ z = s^{2} t \end{array}$

(a) We have to calculate the primary derivatives,

$\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}$

(b) We have to calculate,

$\frac{\partial x}{\partial s}, \frac{\partial y}{\partial s}, \frac{\partial z}{\partial s}$

(c) We have to complete the derivatives $(\frac{\partial f}{\partial s})$ by chain rule.

$\frac{\partial f}{\partial s} = \frac{\partial f}{\partial x} . \frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} . \frac{\partial y}{\partial s} + \frac{\partial f}{\partial z} . \frac{\partial z}{\partial s}$

Answer should be in terms of independent variable s, t.

Answer 6

Chapter 14.6, Problem 1E

To determine

Derivatives of the given function.

$\begin{array}{l} f (x, y, z) = x^{2} y^{3} + z^{4} \\ x = s^{2} \\ y = s t^{2} \\ z = s^{2} t \end{array}$

(a) We have to calculate the primary derivatives,

$\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}$

(b) We have to calculate,

$\frac{\partial x}{\partial s}, \frac{\partial y}{\partial s}, \frac{\partial z}{\partial s}$

(c) We have to complete the derivatives $(\frac{\partial f}{\partial s})$ by chain rule.

$\frac{\partial f}{\partial s} = \frac{\partial f}{\partial x} . \frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} . \frac{\partial y}{\partial s} + \frac{\partial f}{\partial z} . \frac{\partial z}{\partial s}$

Answer should be in terms of independent variable s, t.

Answer 7

Chapter 14.6, Problem 1E

To determine

Derivatives of the given function.

$\begin{array}{l} f (x, y, z) = x^{2} y^{3} + z^{4} \\ x = s^{2} \\ y = s t^{2} \\ z = s^{2} t \end{array}$

(a) We have to calculate the primary derivatives,

$\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}$

(b) We have to calculate,

$\frac{\partial x}{\partial s}, \frac{\partial y}{\partial s}, \frac{\partial z}{\partial s}$

(c) We have to complete the derivatives $(\frac{\partial f}{\partial s})$ by chain rule.

$\frac{\partial f}{\partial s} = \frac{\partial f}{\partial x} . \frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} . \frac{\partial y}{\partial s} + \frac{\partial f}{\partial z} . \frac{\partial z}{\partial s}$

Answer should be in terms of independent variable s, t.

Answer 8

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Answer 9

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Answer 10

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Answer 11

Ch. 14.1 - Prob. 1PQ Ch. 14.1 - Prob. 2PQ Ch. 14.1 - Prob. 3PQ Ch. 14.1 - Prob. 4PQ Ch. 14.1 - Prob. 5PQ Ch. 14.1 - Prob. 1E Ch. 14.1 - Prob. 2E Ch. 14.1 - Prob. 3E Ch. 14.1 - Prob. 4E Ch. 14.1 - Prob. 5E

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Chapter 14 Solutions