Textbook Question
22–36. Derivatives Find the derivatives of the following functions.
f(x) = sinh 4x
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6. Derivatives of Inverse, Exponential, & Logarithmic Functions
Derivatives of Exponential & Logarithmic Functions
1:49 minutesProblem 7.3.27
Textbook Question
22–36. Derivatives Find the derivatives of the following functions.
f(x) = ln sech x
Verified step by step guidance1
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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1mKey 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.
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Derivatives
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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