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

f(x) = ln sech x

Guida verificata passo dopo passo

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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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.

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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.

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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).

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