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

f(x) = ln sech x

Verified step by step guidance

Step 1: Recall the chain rule for derivatives, which states that if a function is composed of two or more functions, the derivative is the derivative of the outer function multiplied by the derivative of the inner function.

Step 2: Identify the outer function and the inner function in f(x) = ln(sech(x)). Here, the outer function is ln(u), and the inner function is u = sech(x).

Step 3: Differentiate the outer function ln(u) with respect to u. The derivative of ln(u) is 1/u. So, d/dx[ln(sech(x))] = (1/sech(x)) * d/dx[sech(x)].

Step 4: Recall the derivative of sech(x). The derivative of sech(x) is -sech(x) * tanh(x). Substitute this into the expression from Step 3.

Step 5: Combine the results. The derivative of f(x) = ln(sech(x)) is (1/sech(x)) * (-sech(x) * tanh(x)). Simplify the expression to get -tanh(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 provides information about the slope of the tangent line to the function's graph 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.

Natural Logarithm (ln)

The natural logarithm, denoted as ln, is the logarithm to the base e, where e is approximately equal to 2.71828. It is a key function in calculus, particularly in relation to exponential growth and decay. The derivative of ln(x) is 1/x, which is essential when differentiating functions that involve natural logarithms, such as ln(sech x) in the given problem.

Hyperbolic Functions

Hyperbolic functions, such as sech (the hyperbolic secant), are analogs of trigonometric functions but are based on hyperbolas rather than circles. The function sech x is defined as 1/cosh x, where cosh x is the hyperbolic cosine function. Understanding the properties and derivatives of hyperbolic functions is crucial for solving problems involving them, including finding the derivative of ln(sech x).

22–36. Derivatives Find the derivatives of the following functions.f(x) = ln sech x