(a) express d w / d t as a function of t, both by using the...

(a) express $d w / d t$ as a function of $t,$ both by using the Chain Rule and by expressing $w$ in terms of $t$ and differentiating directly with respect to $t .$ Then (b) evaluate $d w / d t$ at the given value of $t$ $w=\frac{x}{2}+\frac{y}{z}, \quad x=\cos ^{2} t, \quad y=\sin ^{2} t, \quad z=1 / t ; \quad t=3$

(a) express $d w / d t$ as a function of $t,$ both by using the Chain Rule and by expressing $w$ in terms of $t$ and differentiating directly with respect to $t .$ Then (b) evaluate $d w / d t$ at the given value of $t$
$w=\frac{x}{2}+\frac{y}{z}, \quad x=\cos ^{2} t, \quad y=\sin ^{2} t, \quad z=1 / t ; \quad t=3$

Key Concepts

Chain Rule

The chain rule is a fundamental technique in differential calculus used for computing the derivative of a composite function. When a function is composed of several functions, the chain rule allows you to differentiate the outer function with respect to the inner function and then multiply by the derivative of the inner function with respect to the independent variable.

Direct Differentiation

Direct differentiation involves first rewriting a composite function entirely in terms of the independent variable by substitution. Once the function is expressed explicitly, standard differentiation rules can be applied directly with respect to that variable. This approach can sometimes provide a straightforward alternative to the chain rule.

Composite Functions

Composite functions are functions that are built by applying one function to the results of another. In differentiation, attention must be given to the fact that changes in the innermost variable propagate through the layers of functions. Understanding their structure is crucial for appropriately applying the chain rule or for opting for direct differentiation after substitution.

Evaluation of a Derivative at a Specific Point

Evaluating a derivative at a specific point involves computing the derivative function and then substituting a particular value of the independent variable into the derivative. This process yields the instantaneous rate of change of the original function at that specific point, which is often critical in applications such as optimization and motion analysis.

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