Textbook Question
23–51. Calculating derivatives Find the derivative of the following functions.
y = tan x + cot x
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 3.9.75
75–86. Logarithmic differentiation Use logarithmic differentiation to evaluate f'(x).
f(x) = x^10x
Verified step by step guidance1
Step 1: Start by taking the natural logarithm of both sides of the equation y = f(x) = x^(10x). This gives us ln(y) = ln(x^(10x)).
Step 2: Use the property of logarithms that allows you to bring the exponent down: ln(y) = 10x * ln(x).
Step 3: Differentiate both sides with respect to x. On the left side, use implicit differentiation: (1/y) * (dy/dx). On the right side, apply the product rule to differentiate 10x * ln(x).
Step 4: The product rule states that if u(x) and v(x) are functions of x, then the derivative of their product is u'(x)v(x) + u(x)v'(x). Here, let u(x) = 10x and v(x) = ln(x). Differentiate to find u'(x) = 10 and v'(x) = 1/x.
Step 5: Substitute the derivatives back into the equation from Step 3: (1/y) * (dy/dx) = 10 * ln(x) + 10x * (1/x). Solve for dy/dx by multiplying both sides by y, and then substitute y = x^(10x) back into the equation to express dy/dx in terms of x.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Logarithmic Differentiation
Logarithmic differentiation is a technique 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 function, we can simplify the differentiation process, especially when dealing with complex expressions. This method is particularly useful for functions like f(x) = x^(10x), where both the base and the exponent are variable.
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Logarithmic Differentiation
Chain Rule
The chain rule is a fundamental principle 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 applying logarithmic differentiation, as it allows us to differentiate the logarithm of a function effectively.
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05:02Intro to the Chain Rule
Product Rule
The product rule is a formula used to find the derivative of the product of two functions. It states that if u(x) and v(x) are two differentiable functions, then the derivative of their product is given by u'v + uv'. This rule is particularly relevant in logarithmic differentiation when we encounter products of functions, as it helps in breaking down the differentiation process into manageable parts.
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05:18The Product Rule
Related Practice
Textbook Question
Applying the Chain Rule Use the data in Tables 3.4 and 3.5 of Example 4 to estimate the rate of change in pressure with respect to time experienced by the runner when she is at an altitude of 13,330 ft. Make use of a forward difference quotient when estimating the required derivatives.
Textbook Question
Calculate the derivative of the following functions. In some cases, it is useful to use the properties of logarithms to simplify the functions before computing f'(x).
f(x) = In (2x - 1)(x + 2)³ / (1 - 4x)²
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Textbook Question
Reproduce the graph of f and then plot a graph of f' on the same axes. <IMAGE>
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Textbook Question
The bottom of a large theater screen is 3 ft above your eye level and the top of the screen is 10 ft above your eye level. Assume you walk away from the screen (perpendicular to the screen) at a rate of 3 ft/s while looking at the screen. What is the rate of change of the viewing angle θ when you are 30 ft from the wall on which the screen hangs, assuming the floor is horizontal (see figure)? <IMAGE>
Textbook Question
Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.
g(t) = 6√t
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