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 $t$. $$w=2 y e^{x}-\ln z, \quad x=\ln \left(t^{2}+1\right), \quad y=\tan ^{-1} t, \quad z=e^{t};$ $t=1$$

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 $t$.
$$w=2 y e^{x}-\ln z, \quad x=\ln \left(t^{2}+1\right), \quad y=\tan ^{-1} t, \quad z=e^{t};$ $t=1$$

Key Concepts

Chain Rule

The Chain Rule is a fundamental technique in calculus used to differentiate composite functions. When one variable depends on another and that variable depends on a third variable, the derivative of the composite function is obtained by multiplying the derivative of the outer function evaluated at the inner function by the derivative of the inner function.

Direct Differentiation

Direct Differentiation involves expressing the final function solely in terms of the independent variable before differentiating. This method often simplifies the differentiation process by eliminating intermediate variables and applying standard differentiation rules directly.

Composite Functions

Composite Functions are functions that are formed by combining two or more functions. In differentiation, they often require techniques like the Chain Rule as the variable on which the final function depends is intertwined through several intermediate functions, making the process of finding the derivative more complex.

Product Rule

The Product Rule is used when differentiating an expression that is the product of two functions. It states that the derivative of a product is the derivative of the first function multiplied by the second plus the first function multiplied by the derivative of the second, which is key when dealing with terms like 2y e^x.

Standard Function Derivatives

Understanding Standard Function Derivatives is crucial in calculus because it allows one to differentiate fundamental functions such as exponential, logarithmic, and inverse trigonometric functions quickly. These rules are applied directly once the function is properly decomposed into its basic components.

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