Textbook Question
11–15. Identities Prove each identity using the definitions of the hyperbolic functions.
tanh(−x) = −tanh x
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6. Derivatives of Inverse, Exponential, & Logarithmic Functions
Derivatives of Exponential & Logarithmic Functions
3:47 minutesProblem 7.3.23
Textbook Question
22–36. Derivatives Find the derivatives of the following functions.
f(x) = cosh²x
Verified step by step guidance1
Step 1: Recognize that the function f(x) = cosh²(x) is a composite function. It involves the square of the hyperbolic cosine function, cosh(x). To differentiate it, we will use the chain rule.
Step 2: Recall the chain rule formula: If a function is composed as g(h(x)), then its derivative is g'(h(x)) * h'(x). Here, g(x) = x² and h(x) = cosh(x).
Step 3: Differentiate the outer function g(x) = x². The derivative of x² is 2x. Applying this to g(h(x)), we get 2 * cosh(x).
Step 4: Differentiate the inner function h(x) = cosh(x). Recall that the derivative of cosh(x) is sinh(x).
Step 5: Combine the results from Step 3 and Step 4 using the chain rule. The derivative of f(x) = cosh²(x) is 2 * cosh(x) * sinh(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 its variable. It is a fundamental concept in calculus that allows us to determine how a function behaves at any given point. The derivative can be interpreted as the slope of the tangent line to the graph of the function at a specific point.
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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 is used frequently in calculus, particularly in problems involving exponential growth and decay. Understanding their properties is essential for differentiating functions that involve them.
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Chain Rule
The chain rule is a fundamental technique in calculus used to differentiate 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 dealing with functions like cosh²(x), where one function is nested within another.
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