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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 3.7.25b
Capitolo 3, Problema 3.7.25b

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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b. h′(2)h^{\(\prime\)}\(\left\)(2\(\right\))

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1
Identify that h(x) = f(g(x)) is a composition of functions, which requires the use of the chain rule to find its derivative.
Recall the chain rule: if h(x) = f(g(x)), then h'(x) = f'(g(x)) * g'(x).
To find h'(2), substitute x = 2 into the derivative expression: h'(2) = f'(g(2)) * g'(2).
Use the table to find the values of g(2) and g'(2). Substitute these values into the expression.
Next, use the table to find f'(g(2)) by first finding g(2) and then using this result to find f' at that point. Substitute this value into the expression to complete the calculation of h'(2).

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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(g(x)), the derivative h'(x) can be found by multiplying the derivative of the outer function f with the derivative of the inner function g. Mathematically, this is expressed as h'(x) = f'(g(x)) * g'(x). Understanding this rule is essential for solving problems involving derivatives of composite functions.
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Intro to the Chain Rule

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. In the context of the question, h'(2) specifically refers to the derivative of the function h evaluated at x = 2. Familiarity with this notation is crucial for interpreting and calculating derivatives correctly.
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Sigma Notation

Function Evaluation

Function evaluation involves substituting a specific value into a function to determine its output. In the context of derivatives, evaluating functions at certain points, such as h(2) or g(f(2)), is necessary to compute the derivative using the Chain Rule. This concept is vital for applying the derivatives obtained from the Chain Rule to find specific values, which is often required in calculus problems.
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Evaluating Composed Functions
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