Chapter 12: Q. 31 (page 964)

Use Theorem 12.34 to find the indicated derivatives in Exercises 31–36. Be sure to simplify
$\frac{d x}{d t} w h e n x = ρ \sin φ \cos θ, ρ = t^{2}, φ = t^{3}, a n d θ = t^{4} .$

Short Answer

Expert verified

$The single variable function is, \frac{d x}{d t} = (\sin t^{3} \cos t^{4}) (2 t) + (t^{2} \cos t^{3} \cos t^{4}) (3 t^{2}) + (t^{2} \sin t^{3} (- \sin t^{4})) (4 t^{3}) .$

Step by step solution

Step 1. Given

$x = ρ \sin φ \cos θ, ρ = t^{2}, φ = t^{3}, a n d θ = t^{4} .$

Step 2. Simplification

$Consider the following function, x = ρ \sin φ \cos θ, ρ = t^{2}, φ = t^{3}, a n d θ = t^{4} . Objective is to find \frac{d x}{d t} . By using chain rule, \frac{d x}{d t} = \frac{\partial x}{\partial ρ} . \frac{d ρ}{d t} + \frac{\partial x}{\partial φ} . \frac{d φ}{d t} + \frac{\partial x}{\partial θ} . \frac{d θ}{d t} . \frac{\partial x}{\partial ρ} = \sin φ \cos θ \frac{\partial x}{\partial φ} = ρ \cos φ \cos θ \frac{\partial x}{\partial θ} = ρ \sin φ (- \sin θ) On proceeding the next step, \frac{d ρ}{d t} = 2 t, \frac{d φ}{d t} = 3 t^{2}, \frac{d θ}{d t} = 4 t^{3} . On proceeding the next step, \frac{d x}{d t} = \frac{\partial x}{\partial ρ} . \frac{d ρ}{d t} + \frac{\partial x}{\partial φ} . \frac{d φ}{d t} + \frac{\partial x}{\partial θ} . \frac{d θ}{d t} . \frac{d x}{d t} = (\sin φ \cos θ) (2 t) + (ρ \cos φ \cos θ) (3 t^{2}) + (ρ \sin φ (- \sin θ)) (4 t^{3}) . \frac{d x}{d t} = (\sin t^{3} \cos t^{4}) (2 t) + (t^{2} \cos t^{3} \cos t^{4}) (3 t^{2}) + (t^{2} \sin t^{3} (- \sin t^{4})) (4 t^{3}) . The single variable function is, \frac{d x}{d t} = (\sin t^{3} \cos t^{4}) (2 t) + (t^{2} \cos t^{3} \cos t^{4}) (3 t^{2}) + (t^{2} \sin t^{3} (- \sin t^{4})) (4 t^{3}) .$

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Use Theorem 12.34 to find the indicated derivatives in Exercises 31–36. Be sure to simplify
$\frac{d x}{d t} w h e n x = ρ \sin φ \cos θ, ρ = t^{2}, φ = t^{3}, a n d θ = t^{4} .$

Short Answer

Step by step solution

Step 1. Given

Step 2. Simplification

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Use Theorem 12.34 to find the indicated derivatives in Exercises 31–36. Be sure to simplifydxdtwhenx=ρsinφcosθ,ρ=t2,φ=t3,andθ=t4.

Short Answer

Step by step solution

Step 1. Given

Step 2. Simplification

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

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Discrete Mathematics

Logic and Functions

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Statistics

Study anywhere. Anytime. Across all devices.

Use Theorem 12.34 to find the indicated derivatives in Exercises 31–36. Be sure to simplify
$\frac{d x}{d t} w h e n x = ρ \sin φ \cos θ, ρ = t^{2}, φ = t^{3}, a n d θ = t^{4} .$