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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 3 - DerivativesProblema 3.7.25c
Capitolo 3, Problema 3.7.25c

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

Guida verificata passo dopo passo
1
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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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.
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Intro to the Chain Rule

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.
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Sigma Notation

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.
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Evaluating Composed Functions
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