Find ∂w / ∂s and ∂w / ∂t (a) by using the appropriate Chain...

Find $\partial w / \partial s$ and $\partial w / \partial t$ (a) by using the appropriate Chain Rule and (b) by converting $w$ to a function of $s$ and $t$ before differentiating.
$$w=x^{2}+y^{2}+z^{2}, \quad x=t \sin s, \quad y=t \cos s, \quad z=s t^{2}$$

Key Concepts

Multivariable Chain Rule

The multivariable chain rule is a method used to differentiate composite functions where the dependent variable is a function of intermediate variables that are themselves functions of other independent variables. It allows one to compute the partial derivatives of a composite function with respect to those independent variables by summing the products of the partial derivatives of the outer function and the derivatives of the intermediate functions.

Partial Differentiation

Partial differentiation involves finding the derivative of a function with respect to one variable while keeping the other variables constant. This concept is essential when dealing with functions of several variables, as it isolates the rate of change in one direction without interference from changes in other directions.

Function Composition

Function composition is the process of forming a new function by substituting one function into another. In the context of differentiation, it is important because it establishes the framework where the chain rule applies, enabling the evaluation of derivatives when the variables are interdependent.

Change of Variables

Changing variables involves rewriting a function in terms of a new set of independent variables. This technique can simplify the process of differentiation by expressing the function directly in terms of the new variables, allowing for straightforward application of standard differentiation techniques instead of the chain rule.

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