## Calculus (3rd Edition)

Published by W. H. Freeman

# Chapter 15 - Differentiation in Several Variables - 15.6 The Chain Rule - Preliminary Questions - Page 808: 1

#### Answer

(a) The primary derivatives of $f$ are $$\frac{\partial f}{\partial x}=y, \frac{\partial f}{\partial y}=x.$$ (b) The independent variables are $u,v$.

#### Work Step by Step

(a) The primary derivatives of $f$ are the partial derivatives with respect to $x$ and $y$. We keep all other variables constant while deriving with respect to $x$ and similarly for $y$. Thus, the primary derivatives of $f$ are $$\frac{\partial f}{\partial x}=y, \frac{\partial f}{\partial y}=x.$$ (b) When a function is defined in terms of $x$ and $y$ and the variables $x$ and $y$ depend on other variables, then these other variables are termed independent variables. Thus, in our case, the independent variables are $u,v$.

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