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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3:26 minutes

Problem 7.1.12

Textbook Question

7–28. Derivatives Evaluate the following derivatives.

d/dx (ln³(3x² + 2))

Verified step by step guidance

Step 1: Recognize that the given function is a composite function, specifically the cube of a natural logarithm. Rewrite the function as \( \ln^3(3x^2 + 2) \), which is equivalent to \( [\ln(3x^2 + 2)]^3 \).

Step 2: Apply the chain rule to differentiate \( [\ln(3x^2 + 2)]^3 \). The chain rule states that \( \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) \). Here, \( f(u) = u^3 \) and \( g(x) = \ln(3x^2 + 2) \).

Step 3: Differentiate the outer function \( u^3 \) with respect to \( u \). This gives \( 3u^2 \), so the derivative of the outer function is \( 3[\ln(3x^2 + 2)]^2 \).

Step 4: Now differentiate the inner function \( \ln(3x^2 + 2) \) with respect to \( x \). The derivative of \( \ln(v) \) is \( \frac{1}{v} \cdot v' \), where \( v = 3x^2 + 2 \). Thus, the derivative is \( \frac{1}{3x^2 + 2} \cdot 6x \).

Step 5: Combine the results from Steps 3 and 4. Multiply \( 3[\ln(3x^2 + 2)]^2 \) by \( \frac{6x}{3x^2 + 2} \) to obtain the final derivative expression.

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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 derivative is often denoted as f'(x) or dy/dx, and it can be calculated 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 composition of two functions, say y = f(g(x)), the chain rule states that the derivative dy/dx is the product of the derivative of the outer function f with respect to g and the derivative of the inner function g with respect to x. This is essential for differentiating functions like ln³(3x² + 2).

Natural Logarithm

The natural logarithm, denoted as ln(x), is the logarithm to the base e, where e is approximately 2.71828. It is a key function in calculus, particularly in differentiation and integration. The derivative of ln(x) is 1/x, and when dealing with composite functions involving ln, understanding its properties is crucial for applying the chain rule effectively.

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