In Exercises 1-6,( a express d w / d t as a function of t,...

In Exercises $1-6,($ 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 $t$. $$w=\ln \left(x^{2}+y^{2}+z^{2}\right), \quad x=\cos t, \quad y=\sin t, \quad z=4 \sqrt{t};$ $t=3$$

In Exercises $1-6,($ 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 $t$.
$$w=\ln \left(x^{2}+y^{2}+z^{2}\right), \quad x=\cos t, \quad y=\sin t, \quad z=4 \sqrt{t};$ $t=3$$

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Key Concepts

Chain Rule for Composite Functions

The chain rule is a fundamental tool in calculus used to differentiate compositions of functions. In cases where a function depends on several other functions that themselves are dependent on another variable, the rule allows us to take the derivative of the outer function with respect to each inner variable and then multiply by the derivative of those inner variables with respect to the original variable.

Direct Differentiation

Direct differentiation involves rewriting a composite function as an explicit function of a single variable and then differentiating it normally. This approach bypasses the need to handle intermediate variables separately, which is particularly useful when the composite function can be simplified or expressed in a more straightforward form.

Differentiation of Logarithmic Functions

When differentiating a logarithmic function, such as the natural logarithm, the derivative of ln(u) with respect to its argument u is 1/u. This rule is applied in conjunction with either the chain rule or direct differentiation when the argument of the logarithm is itself a function of another variable.

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