Use Equations 7 to find ∂z / ∂x and ∂z / ∂y. x^2+y^2+z^2=3 x...

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Key Concepts

Implicit Differentiation

Implicit differentiation is a technique used to find derivatives of functions that are not given explicitly in terms of one variable. Instead, the relationship between the variables is provided by an equation. When differentiating such equations, one differentiates both sides with respect to the variable of interest while treating the dependent variables as implicit functions of that variable, and then solves for the desired derivative.

Partial Derivatives

Partial derivatives measure how a multivariable function changes as one variable changes while keeping the other variables constant. They provide the rate of change of the function in the direction of one coordinate axis. This is particularly useful in problems with functions of several variables where changes in each variable have distinct effects on the outcome.

Chain Rule

The chain rule is a fundamental rule in calculus used for differentiating composite functions. In the context of implicit differentiation and partial derivatives, the chain rule allows us to differentiate a function that is composed of one or more functions of other variables. When differentiating with respect to one variable, the chain rule accounts for the effect of that variable on another variable by multiplying by the derivative of the inner function.

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