In Exercises 11 through 14, find the total derivative d u /...

In Exercises 11 through 14, find the total derivative $d u / d t$ by two methods: (a) Use the chain rule; (b) make the substitutions for $x$ and $y$ or for $x, y$, and $z$ before differentiating.
$$
u=y e^{x}+x e^{y} ; x=\cos t ; y=\sin t
$$

Key Concepts

Chain Rule

The chain rule is a fundamental tool in calculus for differentiating composite functions. When a function depends on one or more intermediary variables that are themselves functions of another variable, the chain rule allows you to compute the derivative of the overall function by multiplying the derivative of the outer function with respect to the intermediate variables by the derivatives of these intermediate variables with respect to the final variable.

Total Derivative

The total derivative of a multivariable function with respect to an independent variable is a measure of how the function changes as that variable changes, taking into account both the direct dependence and the indirect impacts through other dependent variables. It effectively sums up the contributions from all paths of dependency from the independent variable to the function.

Partial Derivatives

Partial derivatives are used in multivariable calculus to measure the rate of change of a function with respect to one variable while holding the other variables constant. They are a critical component when applying the chain rule to compute the total derivative of a function that depends on several intermediate variables.

Substitution Method in Differentiation

The substitution method involves replacing the intermediate variables with their expressions in terms of the independent variable before differentiating. This method simplifies the differentiation process by transforming a composite function into a single-variable function, making standard derivative rules directly applicable.

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