Consider the following table of values of the functions f...

Consider the following table of values of the functions f and g and their derivatives at various points. $$\begin{array}{ccccc}{x} & {1} & {2} & {3} & {4} \\ \hline f(x) & {2} & {4} & {1} & {3} \\ {f^{\prime}(x)} & {-6} & {-7} & {-8} & {-9} \\ {g(x)} & {2} & {3} & {4} & {1} \\ {g^{\prime}(x)} & {2 / 7} & {3 / 7} & {4 / 7} & {5 / 7}\end{array}$$ Find the following using the table above. (a) $D_{x}(f[g(x)])$ at $x=1 \quad$ (b) $D_{x}(f[g(x)]) \quad$ at $x=2$

Consider the following table of values of the functions f and g
and their derivatives at various points.

$$\begin{array}{ccccc}{x} & {1} & {2} & {3} & {4} \\ \hline f(x) & {2} & {4} & {1} & {3} \\ {f^{\prime}(x)} & {-6} & {-7} & {-8} & {-9} \\ {g(x)} & {2} & {3} & {4} & {1} \\ {g^{\prime}(x)} & {2 / 7} & {3 / 7} & {4 / 7} & {5 / 7}\end{array}$$
Find the following using the table above.
(a) $D_{x}(f[g(x)])$ at $x=1 \quad$ (b) $D_{x}(f[g(x)]) \quad$ at $x=2$

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Key Concepts

Interpreting Tabulated Values for Functions and Derivatives

When functions and their derivatives are provided in a table, it is important to correctly extract the necessary values when evaluating expressions such as f(g(x)) or applying rules like the chain rule. This concept helps in organizing the given data and substituting the appropriate values into differentiation formulas.

Chain Rule

The chain rule is a fundamental concept in calculus for differentiating composite functions. It states that the derivative of a composite function, f(g(x)), is the product f'(g(x)) multiplied by g'(x). This principle allows one to break down the problem of finding the derivative into two manageable parts: first, differentiating the outer function while keeping the inner function unchanged, and second, multiplying by the derivative of the inner function.

Function Composition

Function composition involves applying one function to the results of another, symbolized as f(g(x)). This concept is critical for understanding how functions interact with each other and is the underlying structure that necessitates the use of the chain rule in differentiation.

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