Assume that the equation defines z implicitly as a function...

Assume that the equation defines $z$ implicitly as a function of $x$ and $y$, and use "implicit partial differentiation" to find $\partial z / \partial x$ and $\partial z / \partial y$.
$$
x-y z+\cos x y z=2
$$

Key Concepts

Chain Rule

The chain rule is a fundamental principle in calculus that is used to differentiate composite functions. In contexts where a variable is a function of others, the chain rule helps compute derivatives by accounting for the dependence of one variable on another, which is pivotal in implicit differentiation.

Implicit Function Theorem

The implicit function theorem provides the theoretical foundation for treating an implicitly defined equation as a function of one or more variables. It establishes the conditions under which equations defining relationships among variables can be solved for one variable as a differentiable function of the others, thereby justifying the use of implicit differentiation.

Implicit Differentiation

Implicit differentiation is a technique used when a function is defined implicitly by an equation rather than explicitly. It involves differentiating both sides of the equation with respect to the independent variable, while considering the dependent variable as a function of that independent variable.

Partial Differentiation

Partial differentiation refers to the process of differentiating a function with several variables with respect to one variable at a time, treating all other variables as constants. This technique is essential in analyzing how a multivariable function changes along each independent direction.

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