Derivatives using tables Let $h\left(x\right)=f\left(g\right(x\left)\right)$ and $p\left(x\right)=g\left(f\right(x\left)\right)$. Use the table to compute the following derivatives.
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a. $h^{\prime }\left(3\right)$

Verified step by step guidance

1. Step 1: Recognize that the function h(x) = f(g(x)) is a composition of functions, and to find its derivative h'(x), we need to use the chain rule.
2. Step 2: The chain rule states that if h(x) = f(g(x)), then h'(x) = f'(g(x)) * g'(x).
3. Step 3: To find h'(3), substitute x = 3 into the expression for h'(x), giving us h'(3) = f'(g(3)) * g'(3).
4. Step 4: Use the table to find the values of g(3), f'(g(3)), and g'(3). First, find g(3) from the table.
5. Step 5: Once you have g(3), use the table to find f'(g(3)) and g'(3), then multiply these values to find h'(3).

Key Concepts

Chain Rule

The Chain Rule is a fundamental theorem in calculus used to differentiate composite functions. It states that if a function h(x) is composed of two functions f and g, such that h(x) = f(g(x)), then the derivative h'(x) can be found using the formula h'(x) = f'(g(x)) * g'(x). This rule is essential for calculating derivatives of functions that are nested within one another.

Derivative Notation

Derivative notation, such as h'(x) or f'(x), represents the rate of change of a function with respect to its variable. It indicates how the function's output changes as its input changes. Understanding this notation is crucial for interpreting and calculating derivatives, especially when dealing with multiple functions and their compositions.

Function Composition

Function composition occurs when one function is applied to the result of another function. For example, if h(x) = f(g(x)), then g(x) is evaluated first, and its output is used as the input for f. This concept is vital for understanding how to differentiate composite functions and apply the Chain Rule effectively.