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.
$$
f(r, \theta)=r^{2} \cos \theta-2 r \tan \theta ; D_{2} f(r, \theta)
$$

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Key Concepts

Differentiation of Trigonometric Functions

Understanding the derivatives of trigonometric functions like cosine and tangent is crucial when they appear in multivariable functions. For example, the derivative of cosine is negative sine, while the derivative of tangent involves the square of the secant, and these identities facilitate the correct computation of partial derivatives involving trigonometric expressions.

Partial Derivatives

Partial derivatives represent the rate at which a multivariable function changes with respect to one variable, while keeping all other variables constant. This concept is fundamental in multivariable calculus and is used to analyze the behavior of functions in multiple dimensions.

Holding Variables Constant

In partial differentiation, all variables except the one of interest are treated as constants. This approach simplifies the computation by reducing the problem to an ordinary differentiation setting, focusing solely on the effect of one variable at a time.

Ordinary Differentiation Rules

Techniques such as the power rule, product rule, and chain rule from single-variable calculus are applied when computing partial derivatives. These rules help in systematically breaking down and calculating the derivative of each term in a multivariable function.

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