Domanda del libro di testo
If f′(x)=3x+2, find the slope of the line tangent to the curve y=f(x) at x=1, 2, and 3.
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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 3.9.27
Find the derivative of the following functions.
y = x² (1 - In x²)
Guida verificata passo dopo passo1
First, identify the function y = x² (1 - ln(x²)). This is a product of two functions: u(x) = x² and v(x) = 1 - ln(x²).
To find the derivative of y, apply the product rule: (u*v)' = u'v + uv'.
Calculate u'(x), the derivative of u(x) = x². Using the power rule, u'(x) = 2x.
Calculate v'(x), the derivative of v(x) = 1 - ln(x²). Use the chain rule: v'(x) = -d/dx[ln(x²)] = -2/x.
Substitute u'(x), v(x), u(x), and v'(x) into the product rule formula: y' = (2x)(1 - ln(x²)) + (x²)(-2/x). Simplify the expression to find the derivative.
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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 a fundamental concept in calculus that provides the slope of the tangent line to the curve at any given point. The derivative is often denoted as f'(x) or dy/dx and can be calculated using various rules, such as the power rule, product rule, and chain rule.
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Percorso guidato
05:44
Derivatives
Product Rule
The product rule is a formula used to find the derivative of the product of two functions. If u(x) and v(x) are two differentiable functions, the product rule states that the derivative of their product is given by u'v + uv'. This rule is essential when differentiating functions that are multiplied together, as seen in the given function y = x²(1 - ln x²).
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05:18The Product Rule
Natural Logarithm
The natural logarithm, denoted as ln, is the logarithm to the base e, where e is approximately equal to 2.71828. It is a key function in calculus, particularly in differentiation and integration. The derivative of ln(x) is 1/x, and understanding how to differentiate functions involving natural logarithms is crucial for solving problems that include ln terms, such as ln(x²) in the given function.
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05:18Derivative of the Natural Logarithmic Function
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