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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2:27 minutes

Problem 7.1.11

Textbook Question

7–28. Derivatives Evaluate the following derivatives.

d/dx ((ln 2x)⁻⁵)

Verified step by step guidance

Step 1: Recognize that the given function is ((ln(2x))⁻⁵). This is a composite function, so we will need to use the chain rule to differentiate it.

Step 2: Apply the chain rule. Let u = ln(2x), so the function becomes u⁻⁵. The derivative of u⁻⁵ with respect to u is -5u⁻⁶.

Step 3: Differentiate u = ln(2x) with respect to x. Using the chain rule again, the derivative of ln(2x) is (1/(2x)) * 2, which simplifies to 1/x.

Step 4: Combine the results from Step 2 and Step 3. Multiply the derivative of u⁻⁵ (-5u⁻⁶) by the derivative of u (1/x). Substitute u = ln(2x) back into the expression.

Step 5: The final derivative is -5(ln(2x))⁻⁶ * (1/x). This is the simplified form of the derivative.

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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. It is a fundamental concept in calculus that allows us to determine how a function behaves as its input changes. The notation d/dx indicates differentiation with respect to the variable x, and derivatives can be computed using various rules, such as the power rule, product rule, and chain rule.

Chain Rule

The chain rule is a formula for computing the derivative of a composite function. If a function y is defined as a function of u, which in turn is a function of x (y = f(u) and u = g(x)), the chain rule states that the derivative dy/dx is the product of the derivative of f with respect to u and the derivative of g with respect to x. This is essential for differentiating functions like (ln(2x))⁻⁵, where the inner function is ln(2x).

Natural Logarithm

The natural logarithm, denoted as ln, is the logarithm to the base e, where e is approximately 2.71828. It is an important function in calculus, particularly in differentiation and integration. The derivative of ln(x) is 1/x, and when dealing with expressions involving ln, understanding its properties and how to differentiate it is crucial for solving problems involving logarithmic functions.