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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2:00 minutes

Problem 7.3.25

Textbook Question

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

f(x) = tanh²x

Verified step by step guidance

Step 1: Recognize that the function f(x) = tanh²(x) is a composite function. It involves the square of the hyperbolic tangent function, tanh(x). To differentiate it, we will use the chain rule.

Step 2: Apply the chain rule. The derivative of tanh²(x) is 2 * tanh(x) * (d/dx of tanh(x)). This is because the outer function is x², and the inner function is tanh(x).

Step 3: Recall the derivative of tanh(x). The derivative of tanh(x) is sech²(x), where sech(x) is the hyperbolic secant function.

Step 4: Substitute the derivative of tanh(x) into the expression from Step 2. This gives: f'(x) = 2 * tanh(x) * sech²(x).

Step 5: Simplify the expression if needed. The derivative of f(x) = tanh²(x) is now fully expressed as f'(x) = 2 * tanh(x) * sech²(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 allows us to determine how a function behaves at any given point. The derivative can be interpreted as the slope of the tangent line to the curve of the function at a specific point.

Chain Rule

The chain rule is a formula for computing the derivative of the composition of two or more functions. It states that if you have a function that is composed of another function, the derivative can be found by multiplying the derivative of the outer function by the derivative of the inner function. This is particularly useful when differentiating functions like f(x) = tanh²x, where tanh is a function itself.

Hyperbolic Functions

Hyperbolic functions, such as tanh, sinh, and cosh, are analogs of the trigonometric functions but are based on hyperbolas instead of circles. The function tanh(x) is defined as the ratio of the hyperbolic sine and cosine, and it has unique properties, including its range and behavior as x approaches infinity. Understanding these functions is essential for differentiating expressions involving them.