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

Previous problemNext problem

1:36 minutes

Problem 3.19

Textbook Question

Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.
y = x⁵

Verified step by step guidance

Step 1: Identify the function for which you need to find the derivative. Here, the function is \( y = x^5 \).

Step 2: Recognize that this is a power function of the form \( y = x^n \), where \( n = 5 \).

Step 3: Apply the power rule for differentiation, which states that the derivative of \( x^n \) is \( nx^{n-1} \).

Step 4: Substitute \( n = 5 \) into the power rule formula to find the derivative: \( \frac{d}{dx}(x^5) = 5x^{5-1} \).

Step 5: Simplify the expression to obtain the derivative: \( 5x^4 \).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Video duration:

Play a video:

Was this helpful?

Key Concepts

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

Derivatives

A derivative represents the rate at which a function changes at a given point. It is a fundamental concept in calculus that measures how a function's output value changes as its input value changes. The derivative can be interpreted as the slope of the tangent line to the graph of the function at a specific 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 functions, making it easier to compute derivatives quickly.

Function Notation

Function notation is a way to represent functions and their outputs. In the context of derivatives, it is common to express a function as y = f(x), where f(x) denotes the function's output for a given input x. Understanding function notation is essential for applying differentiation techniques and interpreting the results correctly.