Textbook Question
29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.
∫ e^{2x} / (4 + e^{2x}) dx
Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
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
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Briggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblem 7.1.10
Briggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblem 7.1.10Chapter 7, Problem 7.1.10
7–28. Derivatives Evaluate the following derivatives.
d/dx (ln (cos² x))
Verified step by step guidance1
Step 1: Recognize that the derivative involves the natural logarithm function ln and the cosine squared function cos²(x). Use the chain rule and logarithmic differentiation to simplify the process.
Step 2: Apply the logarithmic property ln(a^b) = b * ln(a) to rewrite ln(cos²(x)) as 2 * ln(cos(x)). This simplifies the expression and makes differentiation easier.
Step 3: Differentiate the simplified expression 2 * ln(cos(x)) with respect to x. The constant 2 remains unchanged, and you focus on differentiating ln(cos(x)).
Step 4: Use the chain rule to differentiate ln(cos(x)). The derivative of ln(u) is 1/u, and the derivative of cos(x) is -sin(x). Combine these to get (1/cos(x)) * (-sin(x)).
Step 5: Multiply the result from Step 4 by the constant 2 from Step 3. The final derivative is -2 * (sin(x)/cos(x)), which can also be expressed as -2 * tan(x).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivatives
A derivative represents the rate of change of a function with respect to a variable. In calculus, it is a fundamental concept that allows us to determine how a function behaves at any given point. The derivative is often denoted as f'(x) or dy/dx, and it can be interpreted as the slope of the tangent line to the curve of the function at a specific point.
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Derivatives
Chain Rule
The chain rule is a formula for computing the derivative of the composition of two or more functions. It states that if you have a function g(x) that is composed with another function f(u), where u = g(x), then the derivative of f with respect to x can be found by multiplying the derivative of f with respect to u by the derivative of g with respect to x. This is essential for differentiating complex functions like ln(cos² x).
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Logarithmic Differentiation
Logarithmic differentiation is a technique used to differentiate functions that are products or quotients of variables, especially when they involve exponentials or logarithms. By taking the natural logarithm of both sides of an equation, we can simplify the differentiation process. This method is particularly useful for functions like ln(cos² x), as it allows us to apply the properties of logarithms to simplify the expression before differentiating.
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Related Practice
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Textbook Question
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Textbook Question
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2
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