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

3. Techniques of Differentiation

Basic Rules of Differentiation

Calculus

3. Techniques of Differentiation

Basic Rules of Differentiation

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1:46 minutes

Problem 3.9.5

Textbook Question

State the derivative rule for the logarithmic function f(x)=log(subscript b)x. How does it differ from the derivative formula for ln x?

Verified step by step guidance

The derivative of the logarithmic function \( f(x) = \log_b{x} \) is given by the formula \( f'(x) = \frac{1}{x \ln{b}} \). This formula is derived using the change of base formula for logarithms and the chain rule.

To understand this, recall the change of base formula: \( \log_b{x} = \frac{\ln{x}}{\ln{b}} \). This allows us to express the logarithm in terms of the natural logarithm, \( \ln{x} \).

Differentiate \( \frac{\ln{x}}{\ln{b}} \) with respect to \( x \). Since \( \ln{b} \) is a constant, the derivative is \( \frac{1}{\ln{b}} \cdot \frac{d}{dx}(\ln{x}) \).

The derivative of \( \ln{x} \) with respect to \( x \) is \( \frac{1}{x} \). Therefore, the derivative of \( \log_b{x} \) becomes \( \frac{1}{x \ln{b}} \).

In contrast, the derivative of \( \ln{x} \) is simply \( \frac{1}{x} \). The difference arises from the presence of the constant \( \ln{b} \) in the denominator for the derivative of \( \log_b{x} \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Derivative of Logarithmic Functions

The derivative of the logarithmic function f(x) = log_b(x) is given by f'(x) = 1 / (x ln(b)), where b is the base of the logarithm. This rule highlights how the derivative depends on both the input x and the natural logarithm of the base b, indicating that the rate of change of the logarithmic function varies with different bases.

Natural Logarithm

The natural logarithm, denoted as ln(x), is a specific logarithmic function where the base is Euler's number e (approximately 2.718). The derivative of ln(x) is simpler, given by f'(x) = 1/x, which reflects the unique properties of the natural logarithm and its relationship to exponential functions.

Comparison of Derivative Rules

The key difference between the derivatives of log_b(x) and ln(x) lies in the presence of the base b in the former's derivative formula. While ln(x) has a straightforward derivative of 1/x, log_b(x) introduces an additional factor of 1/ln(b), making it essential to consider the base when differentiating logarithmic functions with bases other than e.

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Textbook Question

{Use of Tech} The Witch of Agnesi The graph of y = a³ / x²+a², where a is a constant, is called the witch of Agnesi (named after the 18th-century Italian mathematician Maria Agnesi).

b. Plot the function and the tangent line found in part (a).