Textbook Question
22–36. Derivatives Find the derivatives of the following functions.
f(x) = √coth 3x
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6. Derivatives of Inverse, Exponential, & Logarithmic Functions
Derivatives of Exponential & Logarithmic Functions
4:42 minutesProblem 7.3.34
Textbook Question
22–36. Derivatives Find the derivatives of the following functions.
f(x) = csch⁻¹(2/x)
Verified step by step guidance1
Step 1: Recall the derivative formula for the inverse hyperbolic cosecant function. The derivative of \( \text{csch}^{-1}(u) \) with respect to \( u \) is \( \frac{-1}{|u|\sqrt{u^2 + 1}} \).
Step 2: Identify \( u \) in the given function \( f(x) = \text{csch}^{-1}(\frac{2}{x}) \). Here, \( u = \frac{2}{x} \).
Step 3: Apply the chain rule to find \( f'(x) \). The chain rule states that \( \frac{d}{dx}[\text{csch}^{-1}(u)] = \frac{-1}{|u|\sqrt{u^2 + 1}} \cdot \frac{du}{dx} \).
Step 4: Compute \( \frac{du}{dx} \) for \( u = \frac{2}{x} \). Using the quotient rule, \( \frac{du}{dx} = \frac{d}{dx}[\frac{2}{x}] = -\frac{2}{x^2} \).
Step 5: Substitute \( u = \frac{2}{x} \) and \( \frac{du}{dx} = -\frac{2}{x^2} \) into the chain rule formula to express \( f'(x) \) in terms of \( x \). Simplify the expression without calculating the final value.
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4mKey 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 graph of the function at any given point. The process of finding a derivative is called differentiation, and it involves applying specific rules and formulas to compute the derivative of various types of functions.
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Inverse Hyperbolic Functions
Inverse hyperbolic functions, such as csch⁻¹(x), are the inverses of hyperbolic functions. They are used to solve equations involving hyperbolic functions and have specific derivatives that can be derived from their definitions. Understanding how to differentiate these functions is crucial when working with them, as they often appear in calculus problems involving integration and differentiation.
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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 is composed of two functions u and x (i.e., y = f(u) and u = g(x)), then the derivative of y 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 rule is essential for finding derivatives of functions that involve compositions, such as the function given in the question.
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