Compute the first-order partial derivatives of the function. W = 6x (x² + y² + 2²) 5/2
Compute the first-order partial derivatives of the function. W = 6x (x² + y² + 2²) 5/2
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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![### Compute the First-Order Partial Derivatives of the Function
Given the function:
\[ w = \frac{6x}{\left(x^2 + y^2 + z^2\right)^{5/2}} \]
We need to compute the first-order partial derivatives with respect to \( x \), \( y \), and \( z \). Use symbolic notation and fractions where needed.
---
#### Partial Derivative with Respect to \( x \)
Calculate \( \frac{\partial w}{\partial x} \):
\[
\frac{\partial w}{\partial x} = \frac{6\left(x^2 + y^2 + z^2\right)^{5/2} - 30x^2\left(x^2 + y^2 + z^2\right)^{3/2}}{\left(x^2 + y^2 + z^2\right)^5}
\]
The formula shown is marked incorrect.
---
#### Partial Derivative with Respect to \( y \)
Calculate \( \frac{\partial w}{\partial y} \):
\[
\frac{\partial w}{\partial y} = \frac{-30xy}{\left(x^2 + y^2 + z^2\right)^{7/2}}
\]
---
#### Partial Derivative with Respect to \( z \)
Calculate \( \frac{\partial w}{\partial z} \):
\[
\frac{\partial w}{\partial z} = \frac{-30xz}{\left(x^2 + y^2 + z^2\right)^{7/2}}
\]
---
**Key Points:**
- When calculating the partial derivatives, it is important to apply the product and chain rules accurately.
- For the partial derivative with respect to \( x \), simplify the expression correctly to avoid errors.
- The notation \( \frac{\partial w}{\partial x} \), \( \frac{\partial w}{\partial y} \), and \( \frac{\partial w}{\partial z} \) signifies partial derivatives with respect to \( x \), \( y \), and \( z \) respectively.
This solution illustrates the steps to find the first-order partial derivatives of a function given in a specific form. Perfect accuracy and simplification are critical in obtaining the correct mathematical expressions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4bed6ad9-30ab-4f7a-8335-3ed0abd0b992%2Fd62d5766-0e8a-493a-bc75-6b435ce54bec%2Fbzvsvtd_processed.png&w=3840&q=75)
Transcribed Image Text:### Compute the First-Order Partial Derivatives of the Function
Given the function:
\[ w = \frac{6x}{\left(x^2 + y^2 + z^2\right)^{5/2}} \]
We need to compute the first-order partial derivatives with respect to \( x \), \( y \), and \( z \). Use symbolic notation and fractions where needed.
---
#### Partial Derivative with Respect to \( x \)
Calculate \( \frac{\partial w}{\partial x} \):
\[
\frac{\partial w}{\partial x} = \frac{6\left(x^2 + y^2 + z^2\right)^{5/2} - 30x^2\left(x^2 + y^2 + z^2\right)^{3/2}}{\left(x^2 + y^2 + z^2\right)^5}
\]
The formula shown is marked incorrect.
---
#### Partial Derivative with Respect to \( y \)
Calculate \( \frac{\partial w}{\partial y} \):
\[
\frac{\partial w}{\partial y} = \frac{-30xy}{\left(x^2 + y^2 + z^2\right)^{7/2}}
\]
---
#### Partial Derivative with Respect to \( z \)
Calculate \( \frac{\partial w}{\partial z} \):
\[
\frac{\partial w}{\partial z} = \frac{-30xz}{\left(x^2 + y^2 + z^2\right)^{7/2}}
\]
---
**Key Points:**
- When calculating the partial derivatives, it is important to apply the product and chain rules accurately.
- For the partial derivative with respect to \( x \), simplify the expression correctly to avoid errors.
- The notation \( \frac{\partial w}{\partial x} \), \( \frac{\partial w}{\partial y} \), and \( \frac{\partial w}{\partial z} \) signifies partial derivatives with respect to \( x \), \( y \), and \( z \) respectively.
This solution illustrates the steps to find the first-order partial derivatives of a function given in a specific form. Perfect accuracy and simplification are critical in obtaining the correct mathematical expressions.
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