Chapter 3, Problem 3.15NP

Interpretation Introduction

Interpretation: The value for the following needs be determined for the given function:

  ${\left(\frac{\partial \text{f}}{\partial \text{x}}\right)}_{\text{y}}$ , ${\left(\frac{\partial \text{f}}{\partial \text{y}}\right)}_{\text{x}}$ , ${\left(\frac{{\partial }^2\text{f}}{\partial {\text{x}}^2}\right)}_{\text{y}}$ , ${\left(\frac{{\partial }^2\text{f}}{\partial {\text{y}}^2}\right)}_{\text{x}}$ , ${\left(\frac{\partial }{\partial \text{x}}{\left(\frac{\partial \text{f}}{\partial \text{y}}\right)}_{\text{x}}\right)}_{\text{y}}$ , and ${\left(\frac{\partial }{\partial \text{y}}{\left(\frac{\partial \text{f}}{\partial \text{x}}\right)}_{\text{y}}\right)}_{\text{x}}$ .

Also, whether ${\left(\frac{\partial }{\partial \text{x}}{\left(\frac{\partial \text{f}}{\partial \text{y}}\right)}_{\text{x}}\right)}_{\text{y}}\text{ = }{\left(\frac{\partial }{\partial \text{y}}{\left(\frac{\partial \text{f}}{\partial \text{x}}\right)}_{\text{y}}\right)}_{\text{x}}$ or not needs to be determined

Concept Introduction: The process by which the rate of change of a function or a derivative of a function is determined is said to be differentiation. Some common differential formulas are:

  $\begin{array}{l}\frac{d(x)}{dx}=1\\ \frac{d(x^n)}{dx}=n{x}^{n-1}\\ \frac{d(\sin \text{ x})}{dx}=\cos \text{ x}\\ \frac{d(\cos \text{ x})}{dx}=-\sin \text{ x}\\ \frac{d(a\text{x})}{dx}=a\end{array}$