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. d. p′$$(2)p^{\prime}$$\left(2\right)

Ch. 3 - Derivatives

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Ch. 3 - Derivatives

문제 3.7.25d

3장, 문제 3.7.25d

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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d. $p prime of 2$

검증된 단계별 안내

Identify that you need to find the derivative of the composite function p(x) = g(f(x)) at x = 2, which is p'(2).

Recall the chain rule for derivatives, which states that if you have a composite function p(x) = g(f(x)), then the derivative p'(x) = g'(f(x)) * f'(x).

Evaluate f(x) at x = 2 using the table to find f(2). This will give you the input for g'.

Use the table to find g'(f(2)), which is the derivative of g at the point f(2).

Find f'(2) using the table, which is the derivative of f at x = 2. Multiply g'(f(2)) by f'(2) to get p'(2).

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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'(2), represents the rate of change of a function with respect to its variable. The notation p'(2) specifically indicates the derivative of the function p evaluated at the point x = 2. Understanding this notation is crucial for interpreting and calculating derivatives accurately.

Function Composition

Function composition occurs when one function is applied to the result of another function. In the context of the question, h(x) = f(g(x)) and p(x) = g(f(x)) are examples of composed functions. Recognizing how to work with composed functions is vital for applying the Chain Rule and finding derivatives of such functions.

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>d. p′(2)p^{\(\prime\)}\(\left\)(2\(\right\))