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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1:57 minutes

Problem 7.1.10

Textbook Question

7–28. Derivatives Evaluate the following derivatives.

d/dx (ln (cos² x))

Verified step by step guidance

Step 1: Recognize that the derivative involves the natural logarithm function ln and the cosine squared function cos²(x). Use the chain rule and logarithmic differentiation to simplify the process.

Step 2: Apply the logarithmic property ln(a^b) = b * ln(a) to rewrite ln(cos²(x)) as 2 * ln(cos(x)). This simplifies the expression and makes differentiation easier.

Step 3: Differentiate the simplified expression 2 * ln(cos(x)) with respect to x. The constant 2 remains unchanged, and you focus on differentiating ln(cos(x)).

Step 4: Use the chain rule to differentiate ln(cos(x)). The derivative of ln(u) is 1/u, and the derivative of cos(x) is -sin(x). Combine these to get (1/cos(x)) * (-sin(x)).

Step 5: Multiply the result from Step 4 by the constant 2 from Step 3. The final derivative is -2 * (sin(x)/cos(x)), which can also be expressed as -2 * tan(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 a variable. In calculus, it is a fundamental concept that allows us to determine how a function behaves at any given point. The derivative is often denoted as f'(x) or dy/dx, and it 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 g(x) that is composed with another function f(u), where u = g(x), then the derivative of f with respect to x can be found by multiplying the derivative of f with respect to u by the derivative of g with respect to x. This is essential for differentiating complex functions like ln(cos² x).

Logarithmic Differentiation

Logarithmic differentiation is a technique used to differentiate functions that are products or quotients of variables, especially when they involve exponentials or logarithms. By taking the natural logarithm of both sides of an equation, we can simplify the differentiation process. This method is particularly useful for functions like ln(cos² x), as it allows us to apply the properties of logarithms to simplify the expression before differentiating.

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