In Exercises 15 through 18 , find the total derivative d u /...

In Exercises 15 through 18 , find the total derivative $d u / d t$ by using the chain rule; do not express $u$ as a function of $t$ before differentiating.
$$
u=x y+x z+y z ; x=t \cos t ; y=t \sin t ; z=t
$$

Key Concepts

Chain Rule

The chain rule in multivariable calculus is a method for differentiating a function that depends on one or more other functions, which in turn depend on another variable. It allows for the computation of derivatives of composite functions by multiplying the partial derivatives of the outer function with the derivatives of the inner functions with respect to the independent variable.

Total Derivative

The total derivative measures the overall rate of change of a function with respect to a single variable, accounting for the contributions from all the intermediate variables on which the function depends. It is computed by summing the products of partial derivatives of the function with respect to each variable and the derivative of those variables with respect to the independent variable.

Partial Derivatives

Partial derivatives represent the rate of change of a multivariable function with respect to one variable while keeping the other variables constant. They play a fundamental role in the application of the chain rule for functions of several variables, allowing one to analyze how each independent variable independently contributes to the change in the function.

Parameterized Functions

Parameterized functions are functions in which the variables are expressed in terms of another independent variable, often referred to as a parameter. This concept is essential when applying the chain rule and total derivative, as it involves understanding how each parameterized variable changes with respect to the independent parameter.

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