Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

6. Derivatives of Inverse, Exponential, & Logarithmic Functions

Derivatives of Exponential & Logarithmic Functions

Calculus

6. Derivatives of Inverse, Exponential, & Logarithmic Functions

Derivatives of Exponential & Logarithmic Functions

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1:49 minutes

Problem 7.3.27

Textbook Question

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).