Textbook Question
10–19. Derivatives Find the derivatives of the following functions.
f(t) = cosh t sinh t
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6. Derivatives of Inverse, Exponential, & Logarithmic Functions
Derivatives of Inverse Trigonometric Functions
1:44 minutesProblem 7.R.33b
Textbook Question
Derivatives of hyperbolic functions Compute the following derivatives.
b. d/dx (x sech x)
Verified step by step guidance1
Identify the function to differentiate: \(f(x) = x \cdot \text{sech}(x)\), which is a product of two functions, \(x\) and \(\text{sech}(x)\).
Recall the product rule for derivatives: if \(f(x) = u(x) v(x)\), then \(f'(x) = u'(x) v(x) + u(x) v'(x)\).
Compute the derivative of the first function: \(u(x) = x\), so \(u'(x) = 1\).
Compute the derivative of the second function: \(v(x) = \text{sech}(x)\). Use the fact that \(\frac{d}{dx} \text{sech}(x) = -\text{sech}(x) \tanh(x)\).
Apply the product rule: \(f'(x) = 1 \cdot \text{sech}(x) + x \cdot (-\text{sech}(x) \tanh(x))\), then simplify the expression as needed.
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Video duration:
1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivative of a Product
When differentiating a product of two functions, use the product rule: (fg)' = f'g + fg'. This rule allows you to find the derivative of expressions like x·sech(x) by differentiating each part separately and combining the results.
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The Product Rule
Derivative of the Hyperbolic Secant Function (sech x)
The hyperbolic secant function, sech(x), is defined as 1/cosh(x). Its derivative is -sech(x)·tanh(x), which is essential to apply when differentiating expressions involving sech(x).
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Hyperbolic Functions and Their Properties
Hyperbolic functions like sinh, cosh, and sech have properties similar to trigonometric functions but relate to exponential functions. Understanding their definitions and relationships helps in differentiating and simplifying expressions involving them.
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