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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1:10 minutes

Problem 7.1.14

Textbook Question

7–28. Derivatives Evaluate the following derivatives.

d/dx (x^{π})

Verified step by step guidance

Step 1: Recognize that the given function is x raised to the power of π, where π is a constant. The general formula for differentiating x raised to a constant power is d/dx(x^c) = c * x^(c-1), where c is a constant.

Step 2: Identify the constant c in this problem. Here, c = π, which is a mathematical constant approximately equal to 3.14159.

Step 3: Apply the differentiation formula. Substitute c = π into the formula, resulting in d/dx(x^π) = π * x^(π-1).

Step 4: Simplify the expression. The derivative is now expressed as π * x^(π-1).

Step 5: Conclude that the derivative has been successfully computed symbolically. No further simplification is needed unless numerical evaluation is required.

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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 measures how a function's output value changes as its input value changes. The derivative is often denoted as f'(x) or dy/dx, and it provides critical information about the function's behavior, such as its slope at any given point.

Power Rule

The Power Rule is a basic differentiation rule used to find the derivative of functions in the form of x^n, where n is a real number. According to this rule, the derivative of x^n is n*x^(n-1). This rule simplifies the process of differentiation for polynomial and power functions, making it easier to compute derivatives quickly.

Constant Exponents

In calculus, when dealing with functions that have constant exponents, such as x raised to π (a constant), the Power Rule still applies. The exponent π is treated as a constant, allowing us to differentiate the function using the same principles as with integer exponents. This concept is crucial for understanding how to handle derivatives of functions involving irrational or non-integer exponents.

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