Domanda del libro di testo
The following limits represent f'(a) for some function f and some real number a.
b. Evaluate the limit by computing f'(a).
lim x🠂1 x¹⁰⁰-1 / x-1
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Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 3, Problema 84d
Use the given graphs of f and g to find each derivative. <IMAGE>
d/dx (f(f(x))) |x=4
Guida verificata passo dopo passo1
Step 1: Understand the problem requires finding the derivative of a composite function, specifically f(f(x)), at x = 4.
Step 2: Apply the chain rule for derivatives, which states that the derivative of a composite function f(g(x)) is f'(g(x)) * g'(x).
Step 3: Identify the inner function g(x) as f(x) and the outer function f(g(x)) as f(f(x)).
Step 4: Evaluate the derivative of the outer function f'(f(x)) and the derivative of the inner function f'(x) at x = 4 using the given graphs.
Step 5: Multiply the derivatives from Step 4 according to the chain rule: f'(f(4)) * f'(4).
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Chain Rule
The Chain Rule is a fundamental principle in calculus used to differentiate composite functions. It states that if you have a function f(g(x)), the derivative is f'(g(x)) * g'(x). This rule is essential for finding the derivative of functions where one function is nested inside another, as in the case of f(f(x)).
Video consigliato:
05:02
Intro to the Chain Rule
Derivative
A derivative represents the rate of change of a function with respect to its variable. It is a measure of how a function's output value changes as its input value changes. Understanding how to compute derivatives is crucial for analyzing the behavior of functions, including finding slopes of tangent lines and optimizing functions.
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Percorso guidato
05:44Derivatives
Evaluating Functions
Evaluating functions involves substituting specific values into a function to determine its output. In this context, after finding the derivative of f(f(x)), you will need to evaluate it at x=4. This step is important for obtaining a numerical result that reflects the behavior of the composite function at that particular point.
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Percorso guidato
4:26Evaluating Composed Functions
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