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.
$$
r=e^{-\theta} \cos (\theta+\phi) ; \frac{\partial r}{\partial \theta}
$$

Key Concepts

Differentiation of Trigonometric Functions

Differentiation of trigonometric functions, such as cosine or sine, follows specific rules. For example, the derivative of cosine is negative sine, and when the argument of the trigonometric function is a function itself, the chain rule must be applied to correctly compute the derivative.

Differentiation of Exponential Functions

Differentiating exponential functions involves applying the rules of exponential growth or decay. When the exponent itself is a function of the variable, the chain rule is typically applied, and the derivative incorporates the derivative of the exponent as a multiplicative factor.

Partial Differentiation

Partial differentiation involves taking the derivative of a function with multiple variables with respect to one variable, while treating all other variables as constants. This concept is a fundamental part of multivariable calculus and is crucial for understanding how a change in one variable affects the function independently of the others.

Product Rule

The product rule is used when differentiating a function that is the product of two or more functions of the variable of interest. It states that the derivative of the product is the derivative of the first function times the second function plus the first function times the derivative of the second function.

Chain Rule

The chain rule is a method for differentiating compositions of functions. When a function is nested inside another function, the chain rule provides a systematic way to calculate the derivative by multiplying the derivative of the outer function evaluated at the inner function by the derivative of the inner function.

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