Derivatives using tables Let h(x)=f(g(x))h(x)=f(g(x))h(x)=f(g(x)) and p(x)=g(f(x))p(x)=g(f(x))p(x)=g(f(x)). Use the table to compute the following derivatives. c. p′$$(4)p^{\prime}$$\left(4\right)

Ch. 3 - Derivatives

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Briggs 3rd Edition

Ch. 3 - Derivatives

문제 3.7.25c

3장, 문제 3.7.25c

Derivatives using tables Let $h(x)=f(g(x))$ and $p(x)=g(f(x))$ . Use the table to compute the following derivatives.
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c. $p prime of 4$

검증된 단계별 안내

Identify the function composition for p(x) = g(f(x)). We need to find the derivative p'(x) using the chain rule.

Apply the chain rule for derivatives: If p(x) = g(f(x)), then p'(x) = g'(f(x)) * f'(x).

Evaluate f(x) at x = 4 to find f(4). Use the table to find the value of f(4).

Use the table to find f'(4), the derivative of f at x = 4.

Substitute f(4) into g'(f(x)) to find g'(f(4)) using the table, then multiply g'(f(4)) by f'(4) to find p'(4).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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 each other.

Derivative Notation

Derivative notation, such as f'(x) or p'(4), represents the rate of change of a function at a specific point. The notation p'(4) indicates the derivative of the function p(x) evaluated at x = 4. Understanding this notation is crucial for interpreting the results of differentiation and applying them to specific values.

Function Evaluation

Function evaluation involves substituting a specific value into a function to determine its output. For example, evaluating p(4) means substituting 4 into the function p(x) to find its value. This concept is important when calculating derivatives, as it often requires evaluating the original functions at certain points to find the necessary derivatives for applying the Chain Rule.

Derivatives using tables Let h(x)=f(g(x))h(x)=f(g(x))h(x)=f(g(x)) and p(x)=g(f(x))p(x)=g(f(x))p(x)=g(f(x)). Use the table to compute the following derivatives. <IMAGE>c. p′(4)p^{\(\prime\)}\(\left\)(4\(\right\))