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Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblem 7.3.29
Chapter 7, Problem 7.3.29

22–36. Derivatives Find the derivatives of the following functions.


f(x) = x² cosh² 3x

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Step 1: Recognize that the function f(x) = x² cosh²(3x) is a product of two functions: u(x) = x² and v(x) = cosh²(3x). To find the derivative, apply the product rule: (uv)' = u'v + uv'.
Step 2: Compute the derivative of u(x) = x². Using the power rule, the derivative is u'(x) = 2x.
Step 3: Compute the derivative of v(x) = cosh²(3x). Use the chain rule. Let w(x) = cosh(3x), so v(x) = w²(x). The derivative of w²(x) is 2w(x)w'(x). Then, find w'(x) = sinh(3x) * 3 using the chain rule for cosh(3x). Substitute back to get v'(x) = 2cosh(3x) * sinh(3x) * 3.
Step 4: Substitute u'(x), v(x), u(x), and v'(x) into the product rule formula: f'(x) = u'(x)v(x) + u(x)v'(x). This becomes f'(x) = (2x)(cosh²(3x)) + (x²)(2cosh(3x) * sinh(3x) * 3).
Step 5: Simplify the expression for f'(x) as needed. The derivative is now expressed in terms of x, cosh(3x), and sinh(3x).

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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, and chain rule, depending on the function's structure.
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Derivatives

Product Rule

The product rule is a formula used to find the derivative of the product of two functions. It states that if you have two functions, u(x) and v(x), the derivative of their product is given by u'v + uv'. This rule is essential when differentiating functions that are products of simpler functions, as seen in the given function f(x) = x² cosh² 3x.
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The Product Rule

Hyperbolic Functions

Hyperbolic functions, such as cosh(x), are analogs of trigonometric functions but are based on hyperbolas instead of circles. The function cosh(x) is defined as (e^x + e^(-x))/2 and has unique properties, including its derivatives. Understanding hyperbolic functions is crucial for differentiating expressions involving them, particularly when applying the chain rule in conjunction with the product rule.
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Asymptotes of Hyperbolas
Related Practice
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