Q. 35 Use Theorem 12.34 to find the in... [FREE SOLUTION]

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Chapter 12: Q. 35 (page 961)

Use Theorem 12.34 to find the indicated derivatives in Exercises 31鈥�36. Be sure to simplify your answers.
$\frac{d 蚁}{d t} when 蚁 = \sqrt{x^{2} + y^{2} + z^{2}}, x = \sqrt{t}, y = t^{2}, z = t^{3}$

Short Answer

Expert verified

The value of $\frac{d 蚁}{d t} = \frac{1}{2} (1 + 4 t^{3} + 3 t^{5}) {(t + t^{4} + t^{6})}^{\frac{- 1}{2}}$ .

Step by step solution

Step 1. Given Information.

$蚁 = \sqrt{x^{2} + y^{2} + z^{2}} x = \sqrt{t} y = t^{2} z = t^{3}$

Step 2. Calculation.

By Theorem 12.34, we have

$\frac{d 蚁}{d t} = \frac{鈭� 蚁}{鈭� x} 路 \frac{d x}{d t} + \frac{鈭� 蚁}{鈭� y} 路 \frac{d y}{d t} + \frac{鈭� 蚁}{鈭� z} 路 \frac{d z}{d t} - - - - - - (1)$

So first we find

$\frac{鈭� 蚁}{鈭� x}, \frac{d x}{d t}, \frac{鈭� 蚁}{鈭� y}, \frac{d y}{d t}, \frac{鈭� 蚁}{鈭� z}, \frac{d z}{d t}$

So we have

$\frac{鈭� 蚁}{鈭� x} = \frac{1 路 2 x}{2 \sqrt{x^{2} + y^{2} + z^{2}}} = \frac{x}{\sqrt{x^{2} + y^{2} + z^{2}}} \frac{鈭� 蚁}{鈭� y} = \frac{1 路 2 y}{2 \sqrt{x^{2} + y^{2} + z^{2}}} = \frac{y}{\sqrt{x^{2} + y^{2} + z^{2}}} \frac{鈭� 蚁}{鈭� z} = \frac{1 路 2 z}{2 \sqrt{x^{2} + y^{2} + z^{2}}} = \frac{z}{\sqrt{x^{2} + y^{2} + z^{2}}} \frac{d x}{d t} = \frac{1}{2 t^{\frac{1}{2}}} \frac{d y}{d t} = 2 t \frac{d z}{d t} = 3 t^{2}$

Step 3. Calculation.

Use these above values in (1) we get,

$\frac{d 蚁}{d t} = \frac{鈭� 蚁}{鈭� x} 路 \frac{d x}{d t} + \frac{鈭� 蚁}{鈭� y} 路 \frac{d y}{d t} + \frac{鈭� 蚁}{鈭� z} 路 \frac{d z}{d t} \frac{d 蚁}{d t} = \frac{x}{2 t^{\frac{1}{2}} {(x^{2} + y^{2} + z^{2})}^{\frac{1}{2}}} + \frac{2 y t}{{(x^{2} + y^{2} + z^{2})}^{\frac{1}{2}}} + \frac{3 t^{2} z}{{(x^{2} + y^{2} + z^{2})}^{\frac{1}{2}}}$

So from here, putting the value of $x, y, z$ in terms of $t$ we get

$\frac{d 蚁}{d t} = \frac{x}{2 t^{\frac{1}{2}} {(x^{2} + y^{2} + z^{2})}^{\frac{1}{2}}} + \frac{2 y t}{{(x^{2} + y^{2} + z^{2})}^{\frac{1}{2}}} + \frac{3 t^{2} z}{{(x^{2} + y^{2} + z^{2})}^{\frac{1}{2}}} \frac{d 蚁}{d t} = \frac{1}{\sqrt{t + t^{4} + t^{6}}} (\frac{1}{2} + 2 t^{3} + 3 t^{5}) \frac{d 蚁}{d t} = \frac{1}{2} (1 + 4 t^{3} + 3 t^{5}) {(t + t^{4} + t^{6})}^{\frac{- 1}{2}}$

Step 4. Conclusion.

The value of $\frac{d 蚁}{d t} = \frac{1}{2} (1 + 4 t^{3} + 3 t^{5}) {(t + t^{4} + t^{6})}^{\frac{- 1}{2}}$ .

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91影视

Use Theorem 12.34 to find the indicated derivatives in Exercises 31鈥�36. Be sure to simplify your answers.
$\frac{d 蚁}{d t} when 蚁 = \sqrt{x^{2} + y^{2} + z^{2}}, x = \sqrt{t}, y = t^{2}, z = t^{3}$

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Calculation.

Step 3. Calculation.

Step 4. Conclusion.

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91影视

Use Theorem 12.34 to find the indicated derivatives in Exercises 31鈥�36. Be sure to simplify your answers.d蚁dtwhen蚁=x2+y2+z2,x=t,y=t2,z=t3

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Calculation.

Step 3. Calculation.

Step 4. Conclusion.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Logic and Functions

Pure Maths

Mechanics Maths

Theoretical and Mathematical Physics

Decision Maths

Study anywhere. Anytime. Across all devices.

Use Theorem 12.34 to find the indicated derivatives in Exercises 31鈥�36. Be sure to simplify your answers.
$\frac{d 蚁}{d t} when 蚁 = \sqrt{x^{2} + y^{2} + z^{2}}, x = \sqrt{t}, y = t^{2}, z = t^{3}$