Textbook Question
What are the domain and range of ln x?
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 7.3.22
Briggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblem 7.3.22Chapter 7, Problem 7.3.22
22–36. Derivatives Find the derivatives of the following functions.
f(x) = sinh 4x
Verified step by step guidance1
Step 1: Recall the derivative rule for hyperbolic sine functions. The derivative of sinh(u) with respect to x is cosh(u) * du/dx.
Step 2: Identify the inner function u in the given function f(x) = sinh(4x). Here, u = 4x.
Step 3: Compute the derivative of the inner function u = 4x with respect to x. The derivative of 4x is 4.
Step 4: Apply the chain rule. Substitute u = 4x into the derivative formula: f'(x) = cosh(4x) * 4.
Step 5: Simplify the expression to write the derivative as f'(x) = 4 * cosh(4x).
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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 its variable. It is a fundamental concept in calculus that provides information about the slope of the tangent line to the function's graph at any given point. The derivative can be computed using various rules, such as the power rule, product rule, quotient rule, and chain rule.
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Derivatives
Hyperbolic Functions
Hyperbolic functions, such as sinh (hyperbolic sine) and cosh (hyperbolic cosine), are analogs of the trigonometric functions but are based on hyperbolas instead of circles. The function sinh(x) is defined as (e^x - e^(-x))/2. Understanding these functions is crucial for differentiating expressions involving them, as they have specific derivatives that differ from their trigonometric counterparts.
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Chain Rule
The chain rule is a fundamental technique for finding the derivative of composite functions. It states that if a function y = f(g(x)) is composed of two functions, the derivative can be found by multiplying the derivative of the outer function f with the derivative of the inner function g. This rule is particularly useful when differentiating functions like f(x) = sinh(4x), where the inner function is 4x.
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05:02Intro to the Chain Rule
Related Practice
Textbook Question
22–36. Derivatives Find the derivatives of the following functions.
f(x) = csch⁻¹(2/x)
Textbook Question
Bounds on e Use a left Riemann sum with at least n = 2 subintervals of equal length to approximate ln 2 = ∫[1 to 2] (dt/t) and show that ln 2 < 1. Use a right Riemann sum with n = 7 subintervals of equal length to approximate ln 3 = ∫[1 to 3] (dt/t) and show that ln 3 > 1.
Textbook Question
Evaluate ∫ 4ˣ dx.
Textbook Question
Sketch the graphs of y = cosh x, y = sinh x, and y = tanh x (include asymptotes), and state whether each function is even, odd, or neither.
Textbook Question
On what interval is the formula d/dx (tanh⁻¹ x) = 1/(1 - x²) valid?