Q. 31 Use Theorem 12.34 to find the in... [FREE SOLUTION]

Chapter 12: Q. 31 (page 964)

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

Short Answer

Expert verified

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

Step by step solution

Step 1. Given

$x = 蚁 \sin 蠁 \cos 胃, 蚁 = t^{2}, 蠁 = t^{3}, a n d 胃 = t^{4} .$

Step 2. Simplification

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

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

Short Answer

Step by step solution

Step 1. Given

Step 2. Simplification

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91影视

Use Theorem 12.34 to find the indicated derivatives in Exercises 31鈥�36. Be sure to simplifydxdtwhenx=蚁sin蠁cos胃,蚁=t2,蠁=t3,and胃=t4.

Short Answer

Step by step solution

Step 1. Given

Step 2. Simplification

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Applied Mathematics

Statistics

Discrete Mathematics

Theoretical and Mathematical Physics

Probability and Statistics

Study anywhere. Anytime. Across all devices.

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