Professional Courses
Industry-relevant training in Business, Technology, and Design to help professionals and graduates upskill for real-world careers.
Categories
Interactive Games
Fun, engaging games to boost memory, math fluency, typing speed, and English skills—perfect for learners of all ages.
Typing
Memory
Math
English Adventures
Knowledge
Enroll to start learning
You’ve not yet enrolled in this course. Please enroll for free to listen to audio lessons, classroom podcasts and take mock test.
Interactive Audio Lesson
Listen to a student-teacher conversation explaining the topic in a relatable way.
Finding the Derivative Using Basic Rules
Unlock Audio Lesson
Signup and Enroll to the course for listening the Audio Lesson
Today, we’re going to learn how to find derivatives using simple rules. Let's start with the power rule. The power rule states that for any function represented as 𝑓(𝑥) = 𝑥^n, the derivative is 𝑓′(𝑥) = n𝑥^(n-1). Can anyone tell me what this means?
It means we multiply the power by the coefficient and then reduce the power by one?
Exactly! Now, let’s apply this to our first example: finding the derivative of 𝑓(𝑥) = 5𝑥^4 − 3𝑥^2 + 2. Who wants to try it?
I can try! For 5𝑥^4, the derivative is 20𝑥^3, and for -3𝑥^2, it is -6𝑥.
And the constant 2 drops out because its derivative is zero.
Great job! So, what’s the final derivative?
It's 20𝑥^3 − 6𝑥.
Perfect! That’s our derivative. Remember, when finding derivatives, just follow the rules like a recipe. Next, let's move on to what the slope means in a real context.
Finding the Slope of the Tangent Line
Unlock Audio Lesson
Signup and Enroll to the course for listening the Audio Lesson
Now, let’s find the slope of the tangent line to the function y = 𝑥^3 − 2𝑥 at the point where x = 1. Who can start us off?
First, we need to find the derivative, right? So, y' = 3𝑥^2 − 2.
Exactly! Now, let’s evaluate this derivative at x = 1. What do we get?
y'(1) = 3(1)^2 − 2, which is 3 − 2 = 1.
You got it! The slope of the tangent line at that point is 1. Now, how do we write the equation of the tangent line?
We use the point-slope form: y - f(a) = f′(a)(x - a). Here, f(1) is -1.
Right! So, plug that in with the slope. What do we have?
The equation is y + 1 = 1(x − 1), simplifying to y = x − 2.
Excellent! You all are really grasping this concept well. Remember the steps to find derivatives and how to apply them to find slopes and tangent line equations.
Understanding the Definition of a Derivative
Unlock Audio Lesson
Signup and Enroll to the course for listening the Audio Lesson
Let’s discuss the limit definition of a derivative. It’s important for understanding how derivatives truly work. The formula is f′(a) = lim(h→0) [f(a + h) - f(a)]/h. Can anyone explain what this means?
It shows that a derivative is essentially the slope of the secant line as h gets really small, right?
Exactly! As h approaches zero, the secant line gets closer to the tangent line. Let’s see how to find the derivative using this limit definition for f(x) = x^2. Who wants to give it a shot?
Okay, so f′(a) = lim(h→0) [(a + h)^2 - a^2]/h. Simplifying that gives us lim(h→0) [(2ah + h^2)/h].
Then we can cancel h to get lim(h→0) [2a + h]. As h approaches zero, it just equals 2a.
Fantastic! So what’s the derivative of x^2 then?
It’s f′(x) = 2x!
Great work, everyone! Understanding this definition deepens our grasp of what a derivative really calculates.
Introduction & Overview
Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.
Quick Overview
Standard
Utilizing the power rule, sum rule, and derivative definition, this section guides students through a series of examples that demonstrate how to compute derivatives of different functions. Additionally, it clarifies how to find the slope of the tangent line at specific points and formulates equations for these lines.
Detailed
In this section, we delve into practical applications of derivatives by analyzing specific examples. Understanding the steps involved in finding the derivative of functions enables students to grasp the concept of the instantaneous rate of change more effectively. Whether applying the power rule to differentiate polynomial functions or finding the equation of a tangent line, these step-by-step methodologies create a comprehensive path towards mastering derivatives. Key examples illustrate the utility of derivatives in determining slopes and deriving tangent line equations, thereby laying a solid foundation for students in their mathematical journey.
Audio Book
Dive deep into the subject with an immersive audiobook experience.
Example 1: Finding the Derivative
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
Example 1: Find the derivative of 𝑓(𝑥) = 5𝑥^4 − 3𝑥^2 + 2.
Using the power rule and sum rule:
𝑓′(𝑥) = 5⋅4𝑥^3 − 3⋅2𝑥^1 + 0 = 20𝑥^3 − 6𝑥.
Detailed Explanation
In this example, we are tasked with finding the derivative of the function f(x) = 5x^4 - 3x^2 + 2. We apply two differentiation rules: the power rule and the sum rule.
- Power Rule: This rule states that if you have a term in the form of x^n, the derivative is n * x^(n-1). We apply this rule to each term of the polynomial.
- For 5x^4, we multiply 5 by the exponent 4 which equals 20, and then reduce the exponent by 1, giving us 20x^3.
- For -3x^2, we multiply -3 by 2 giving us -6 and reduce the exponent, resulting in -6x.
- The term +2 is a constant, and its derivative is 0.
- We then combine these results to get the complete derivative: f′(x) = 20x^3 - 6x.
Examples & Analogies
Think of this process like a factory producing two different products (the terms 5x^4 and -3x^2) with different production rates. The power rule helps us understand how fast each product quantity is changing as we change the number of workers (x). For each adjustment in workforce, we determine the new rate of production (the derivative) for overall efficiency.
Example 2: Slope of the Tangent Line
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
Example 2: Find the slope of the tangent line to 𝑦 = 𝑥^3 − 2𝑥 at 𝑥 = 1.
1. Find derivative:
𝑦′ = 3𝑥^2 − 2.
2. Evaluate at 𝑥 = 1:
𝑦′(1) = 3(1)^2−2 = 3−2 = 1.
The slope is 1.
3. Find 𝑦 at 𝑥 = 1:
𝑦(1) = 1−2 = −1.
4. Equation of tangent line:
𝑦−(−1) = 1(𝑥−1) ⟹ 𝑦+1 = 𝑥−1 ⟹ 𝑦 = 𝑥 − 2.
Detailed Explanation
In this example, we want to find the slope of the tangent line to the curve defined by y = x^3 - 2x at the specific point where x = 1.
- Step 1: Differentiate the Function: Start by finding the derivative of the function y = x^3 - 2x, which tells us how steep the curve is at any point. Using the power rule:
- The derivative is y' = 3x^2 - 2.
- Step 2: Evaluate the Derivative at x = 1: Substitute x = 1 into the derivative to find the slope of the tangent line at that point:
- y'(1) = 3(1)^2 - 2 = 3 - 2 = 1. The slope of the tangent line is 1.
- Step 3: Find the y-value at x = 1: We need the point on the curve where x = 1. Substitute x = 1 into the original function:
- y(1) = 1 - 2 = -1. So the point is (1, -1).
- Step 4: Write the Equation of the Tangent Line: Now that we have the slope (1) and a point (1, -1), we can write the equation of the tangent line using the point-slope formula y - y₁ = m(x - x₁):
- y - (-1) = 1(x - 1) simplifies to y = x - 2.
Examples & Analogies
Imagine you're driving along a curvy road represented by the function y = x^3 - 2x. You want to know how steep the road is at a specific point (x = 1). The slope of the tangent line gives you the current incline of the road right beneath your car, similar to using a spirit level to see the angle of a slope. The steps we took to find that slope reflect figuring out how steep your incline is based on the curvature of the road.
Example 3: Derivative Using the Definition
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
Example 3: Find the derivative using the definition for 𝑓(𝑥) = 𝑥^2.
(𝑎 + ℎ)^2 − 𝑎^2 𝑎^2 + 2𝑎ℎ + ℎ^2 - 𝑎^2
𝑓′(𝑎) = lim = lim = lim = lim(2𝑎 + ℎ) = 2𝑎.
ℎ→0 ℎ ℎ→0 ℎ ℎ→0
Thus, 𝑓′(𝑥) = 2𝑥.
Detailed Explanation
In this example, we demonstrate how to find the derivative of f(x) = x^2 using its definition based on limits.
- Understanding the Limit Definition: The derivative at a point a is defined as:
- f′(a) = lim (f(a + h) - f(a)) / h as h approaches 0. Here, h is a very small number that represents a change in x.
- Step 1: Write the Difference Quotient: First, substitute f(x) into the definition:
- f′(a) = lim ((a + h)^2 - a^2) / h.
- Step 2: Expand the Numerator: Expanding (a + h)^2 results in a^2 + 2ah + h^2, so:
- f′(a) = lim (a^2 + 2ah + h^2 - a^2) / h, which simplifies to lim (2ah + h^2) / h.
- Step 3: Simplify the Expression: Factor out h from the numerator:
- f′(a) = lim h(2a + h) / h. The h cancels out:
- f′(a) = lim (2a + h) as h approaches 0. This simplifies to 2a.
- Conclusion: Thus, we find that the derivative of f(x) = x^2 is f′(x) = 2x.
Examples & Analogies
Picture a ball rolling down a hill; you want to find out how steep the hill is at a certain point (the slope). To do this, you first observe just a tiny section of the hill right around that point. By analyzing this small segment (with h being a tiny movement up or down the hill), you determine that the slope at that point is how fast the ball is rolling at that exact moment. This captures the essence of determining the derivative: finding the slope right where you are, even with a slight movement to better understand the curve.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
-
Derivative: Represents the instantaneous rate of change or slope of a function.
-
Limit Definition: Derivative defined as the limit of the difference quotient as h approaches zero.
-
Power Rule: A method for finding derivatives of polynomial functions.
-
Tangent Line: The line that best approximates a function at a given point.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
-
Example 1: Differentiate 𝑓(𝑥) = 5𝑥^4 − 3𝑥^2 + 2 to get 𝑓′(𝑥) = 20𝑥^3 − 6𝑥.
-
Example 2: Find the slope of the tangent line to 𝑦 = 𝑥^3 − 2𝑥 at 𝑥 = 1, resulting in the equation 𝑦 = 𝑥 − 2.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
🎵 Rhymes Time
-
Power rule, oh what a treat, multiply and reduce, can't be beat!
📖 Fascinating Stories
-
Imagine a runner on a track. To know how fast they're going at any moment, we look at their position function and find the derivative – that gives us their speed, just like finding the slope of a tangent line!
🧠 Other Memory Gems
-
To remember the steps for derivatives, think 'PST' - Power, Sum, Tangent.
🎯 Super Acronyms
Use 'DERS' for finding derivatives
- Differentiate
- Evaluate
- Repeat
- Simplify.
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
-
Term: Derivative
Definition:
A measure of how a function changes as its input changes, representing the slope of the tangent line.
-
Term: Power Rule
Definition:
A shortcut for finding the derivative of power functions, stating that f'(x) = n*x^(n-1) for f(x) = x^n.
-
Term: Tangent Line
Definition:
A straight line touching a curve at a specific point without crossing it, representing the instantaneous slope of the function at that point.
-
Term: Sum Rule
Definition:
A rule that states the derivative of a sum of functions is the sum of their derivatives.
-
Term: Difference Quotient
Definition:
The expression (f(a + h) - f(a))/h used to define the derivative.