### Computation of First-Order Partial Derivatives #### Function The given function is: \[ P = e^{\sqrt[8]{y^8 + z^8}} \] #### Objective Compute the first-order partial derivatives of the given function. #### Solution 1. **Partial derivative with respect to \( y \):** The first-order partial derivative of \( P \) with respect to \( y \) is given by: \[ \frac{\partial P}{\partial y} = y \cdot \frac{y^7}{\sqrt[8]{y^8 + z^8}} \cdot e^{\sqrt[8]{y^8 + z^8}} \] 2. **Partial derivative with respect to \( z \):** The first-order partial derivative of \( P \) with respect to \( z \) is given by: \[ \frac{\partial P}{\partial z} = \frac{4z^7}{y^8 + z^8} \cdot e^{\sqrt[8]{y^8 + z^8}} \]

### Computation of First-Order Partial Derivatives #### Function The given function is: \[ P = e^{\sqrt[8]{y^8 + z^8}} \] #### Objective Compute the first-order partial derivatives of the given function. #### Solution 1. Partial derivative with respect to \( y \): The first-order partial derivative of \( P \) with respect to \( y \) is given by: \[ \frac{\partial P}{\partial y} = y \cdot \frac{y^7}{\sqrt[8]{y^8 + z^8}} \cdot e^{\sqrt[8]{y^8 + z^8}} \] 2. Partial derivative with respect to \( z \): The first-order partial derivative of \( P \) with respect to \( z \) is given by: \[ \frac{\partial P}{\partial z} = \frac{4z^7}{y^8 + z^8} \cdot e^{\sqrt[8]{y^8 + z^8}} \]

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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