Domanda del libro di testo
Evaluate and simplify y'.
y = 2x√2
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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 3.R.65
Find f′(1) when f(x) = x^(1/x).
Guida verificata passo dopo passo1
First, recognize that the function f(x) = x^(1/x) can be rewritten using the exponential and logarithmic functions for easier differentiation. Express it as f(x) = e^(ln(x^(1/x))).
Simplify the expression using the property of logarithms: ln(x^(1/x)) = (1/x) * ln(x). Therefore, f(x) = e^((1/x) * ln(x)).
To differentiate f(x), use the chain rule. Let u(x) = (1/x) * ln(x), so f(x) = e^(u(x)). The derivative f'(x) = e^(u(x)) * u'(x).
Find u'(x) by differentiating u(x) = (1/x) * ln(x). Use the product rule: u'(x) = d/dx(1/x) * ln(x) + (1/x) * d/dx(ln(x)).
Calculate the derivatives: d/dx(1/x) = -1/x^2 and d/dx(ln(x)) = 1/x. Substitute these into the expression for u'(x) and simplify. Finally, substitute back into f'(x) = e^(u(x)) * u'(x) and evaluate at x = 1 to find f'(1).
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Derivative
The derivative of a function measures how the function's output value changes as its input value changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. In this context, finding f′(1) involves calculating the derivative of f(x) at the specific point x = 1.
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Percorso guidato
05:44
Derivatives
Chain Rule
The chain rule is a fundamental technique in calculus used to differentiate composite functions. It states that if a function y = g(u) is composed with u = f(x), then the derivative dy/dx can be found by multiplying the derivative of g with respect to u by the derivative of f with respect to x. This rule is essential when dealing with functions like f(x) = x^(1/x), which can be expressed in a composite form.
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05:02Intro to the Chain Rule
Logarithmic Differentiation
Logarithmic differentiation is a method used to differentiate functions that are products or quotients of variables raised to variable powers. By taking the natural logarithm of both sides of the equation, we can simplify the differentiation process. This technique is particularly useful for functions like f(x) = x^(1/x), allowing us to handle the exponent more easily when finding the derivative.
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06:30Logarithmic Differentiation
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