In Exercises 13 through 24 , find the indicated partial...

In Exercises 13 through 24 , find the indicated partial derivatives by holding all but one of the variables constant and applying theorems for ordinary differentiation. $$ u=\left(x^{2}+y^{2}+z^{2}\right)^{-1 / 2} ; \frac{\partial u}{\partial z} $$ 20. $u=\tan ^{-1}(x y z w) ; \frac{\partial u}{\partial w}$

In Exercises 13 through 24 , find the indicated partial derivatives by holding all but one of the variables constant and applying theorems for ordinary differentiation.
$$
u=\left(x^{2}+y^{2}+z^{2}\right)^{-1 / 2} ; \frac{\partial u}{\partial z}
$$
20. $u=\tan ^{-1}(x y z w) ; \frac{\partial u}{\partial w}$

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Key Concepts

Partial Differentiation

Partial differentiation is the process of finding the derivative of a multivariable function with respect to one variable while keeping the other variables constant. This technique is essential for understanding how changes in one independent variable affect the dependent variable in functions that depend on multiple variables.

Chain Rule for Multivariable Functions

The chain rule is a fundamental tool in calculus used to differentiate composite functions. In the context of partial differentiation, it allows you to compute the derivative of a function that is composed of another function of multiple variables by differentiating the outer function with respect to the inner function and then multiplying by the derivative of the inner function with respect to the chosen variable.

Differentiation of Inverse Trigonometric Functions

Differentiating inverse trigonometric functions involves applying specific formulas, such as the derivative of arctan(u) which is 1/(1+u^2) multiplied by the derivative of u. This concept is crucial when the function to be differentiated is an inverse trigonometric function of another function, requiring a combination of standard differentiation rules and the chain rule.

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