Textbook Question
9–61. Evaluate and simplify y'.
y = 10^sin x+sin¹⁰x
Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- 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: 9780136847243Not the one you use?Change textbook
Chapter 3, Problem 3.R.10
Evaluate and simplify y'.
y = 4x⁴ ln x − x⁴
Verified step by step guidance1
First, identify the given expression: y'.y = 4x⁴ ln x − x⁴. This implies that y' is the derivative of y with respect to x.
To solve for y', we need to isolate y' in the equation. Start by dividing both sides of the equation by y, giving us y' = (4x⁴ ln x − x⁴) / y.
Next, consider simplifying the expression on the right-hand side. Notice that both terms in the numerator have a common factor of x⁴. Factor out x⁴ to get y' = x⁴(4 ln x − 1) / y.
Now, if y is a function of x, you might need to express y in terms of x to further simplify or evaluate y'. This depends on additional information about y.
Finally, if y is known or can be expressed in terms of x, substitute it into the equation to find y' explicitly. If not, y' is expressed in terms of y and x as derived.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Differentiation
Differentiation is the process of finding the derivative of a function, which represents the rate of change of the function with respect to its variable. In this context, we need to apply differentiation rules to the given function y = 4x⁴ ln x − x⁴ to find y'. This involves using the product rule for the term 4x⁴ ln x and the power rule for the term -x⁴.
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Product Rule
The product rule is a formula used to differentiate products 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'. In this problem, we will apply the product rule to differentiate the term 4x⁴ ln x, where u = 4x⁴ and v = ln x.
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Logarithmic Differentiation
Logarithmic differentiation is a technique used to differentiate functions involving logarithms, which can simplify the differentiation process. In this case, the term ln x in the function y = 4x⁴ ln x − x⁴ requires us to recognize how the logarithmic properties interact with the polynomial term. Understanding how to differentiate ln x is essential for correctly applying the product rule.
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Related Practice
Textbook Question
Find f′(1) when f(x) = x^(1/x).
Textbook Question
Evaluate and simplify y'.
y = 4u²+u / 8u+1
Textbook Question
9–61. Evaluate and simplify y'.
y = 5t² sin t
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Textbook Question
Find the derivative of the inverse of the following functions. Express the result with x as the independent variable.
f(x) = x^-1/3
Textbook Question
The population of the United States (in millions) by decade is given in the table, where t is the number of years after 1910. These data are plotted and fitted with a smooth curve y = p(t) in the figure. <IMAGE><IMAGE>
Compute the average rate of population growth from 1950 to 1960.