3.2 - Derivative Rules
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Power Rule
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Initially, let’s discuss the Power Rule. If we have a function of the form f(x) = x^n, who can tell me what the derivative looks like?
Is it something like d/dx[x^n] = n * x^(n-1)?
Exactly! So, if f(x) = x^3, can anyone find f'(x)?
That would be 3x^2!
Great job! A mnemonic to remember this is 'Drop and Multiply', where you drop the exponent and multiply it by the base reduced by one. Let’s move to the next rule.
Sum Rule
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Now, let’s look at the Sum Rule. When you differentiate a sum of functions, what happens?
We just differentiate each function separately and add the results?
Exactly! d/dx[f(x) + g(x)] = f'(x) + g'(x). Can you apply this to f(x) = x^2 + 3x?
So, f'(x) would be 2x + 3!
Right again! Remember, just differentiate each part separately. Let’s advance to the Product Rule.
Product Rule
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Next is the Product Rule. When differentiating a product of two functions, what should we do?
You differentiate the first function and multiply it by the second function, and then add the first function multiplied by the derivative of the second?
Exactly, it’s g'(x)h(x) + g(x)h'(x). Can anyone give me an example?
For f(x) = x^2 * sin(x), f'(x) would be 2x * sin(x) + x^2 * cos(x)!
Correct! Always remember to apply both functions in that formula. Let’s explore the Quotient Rule next.
Quotient Rule
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The Quotient Rule is next! Can someone detail how we differentiate a function that is a quotient?
We use (g'(x)h(x) - g(x)h'(x)) / h(x)^2?
Fantastic! Suppose we have f(x) = x^2/(x + 1). What’s the derivative?
That would be [2x(x + 1) - x^2(1)] / (x + 1)^2!
Exactly! Just remember, it’s crucial to keep the denominator squared in the result.
Chain Rule
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Lastly, let’s discuss the Chain Rule. What’s the process when differentiating composite functions?
You take the derivative of the outer function and multiply it by the derivative of the inner function?
Correct! d/dx[f(g(x))] = f'(g(x))g'(x). Can anyone provide an example?
For f(x) = sin(x^2), the derivative would be cos(x^2) * 2x!
Excellent! Always make sure to identify both layers of your composite function. Let’s recap what we learned today!
Introduction & Overview
Read summaries of the section's main ideas at different levels of detail.
Quick Overview
Standard
The derivative rules outlined in this section, including the Power Rule, Sum Rule, Product Rule, Quotient Rule, and Chain Rule, are essential tools for calculating the derivatives of different functions, providing crucial insight into how functions change.
Detailed
Derivative Rules
In calculus, understanding how to differentiate functions is vital. This section introduces several primary derivative rules that streamline the process of finding derivatives for various functions. The five key rules discussed are:
Differentiation Rules
1. Power Rule
-
Rule:
\[
\frac{d}{dx}[x^n] = n \cdot x^{n-1}
\] -
Example:
For \( f(x) = x^3 \),
\[
f'(x) = 3x^2
\] -
Self-Check Problem:
Differentiate \( f(x) = x^5 \).
2. Sum Rule
-
Rule:
\[
\frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x)
\] -
Example:
For \( f(x) = x^2 + 3x \),
\[
f'(x) = 2x + 3
\] -
Self-Check Problem:
Differentiate \( f(x) = x^3 + 2x^2 + 7 \).
3. Product Rule
-
Rule:
If \( f(x) = g(x) \cdot h(x) \), then
\[
\frac{d}{dx}[f(x)] = g'(x)h(x) + g(x)h'(x)
\] -
Example:
For \( f(x) = x^2 \cdot \sin x \),
\[
f'(x) = 2x \cdot \sin x + x^2 \cdot \cos x
\] -
Self-Check Problem:
Differentiate \( f(x) = (x^3)(e^x) \).
4. Quotient Rule
-
Rule:
If \( f(x) = \dfrac{g(x)}{h(x)} \), then
\[
\frac{d}{dx}[f(x)] = \frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2}
\] -
Example:
For \( f(x) = \dfrac{x^2}{x+1} \),
\[
f'(x) = \frac{2x(x+1) - x^2(1)}{(x+1)^2}
\] -
Self-Check Problem:
Differentiate \( f(x) = \dfrac{\sin x}{x} \).
5. Chain Rule
-
Rule:
If \( f(x) = F(g(x)) \), then
\[
\frac{d}{dx}[f(x)] = F'(g(x)) \cdot g'(x)
\] -
Example:
For \( f(x) = \sin(x^2) \),
\[
f'(x) = \cos(x^2) \cdot 2x
\] -
Self-Check Problem:
Differentiate \( f(x) = e^{3x^2} \).
These rules provide a structured approach to differentiation and are fundamental in solving various calculus problems.
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Power Rule
Chapter 1 of 5
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Chapter Content
If 𝑓(𝑥) = 𝑥𝑛, where 𝑛 is a constant, then
\[ \frac{d}{dx}[x^n] = n x^{n-1} \]
Example: 𝑑 [𝑥³] = 3𝑥².
Detailed Explanation
The Power Rule is a fundamental rule in differentiation that explains how to find the derivative of power functions. If a function is in the form of 𝑓(𝑥) = 𝑥 raised to some constant power 𝑛, the derivative is calculated by multiplying the original exponent 𝑛 by the base 𝑥 raised to the exponent reduced by 1. For instance, if you have x^3, applying this rule gives you 3x² (where you multiply by 3 and decrease the exponent from 3 to 2).
Examples & Analogies
Imagine a box with a square base whose area you want to maximize. If the side of the base is 𝑥, then the area is 𝑓(𝑥) = 𝑥². To find how fast the area changes as you slightly change the side length, you'd use the Power Rule. If you differentiate, you'll find out how sensitive the area is to changes in the side length.
Sum Rule
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Chapter Content
If 𝑓(𝑥) = 𝑔(𝑥) + ℎ(𝑥), then
\[ \frac{d}{dx}[f(x)] = \frac{d}{dx}[g(x)] + \frac{d}{dx}[h(x)] \]
Example: 𝑑 [𝑥² + 3𝑥] = 2𝑥 + 3.
Detailed Explanation
The Sum Rule states that the derivative of a sum of functions is the sum of their derivatives. This means if you have two functions 𝑔(𝑥) and ℎ(𝑥), and you want to find the derivative of their sum, all you need to do is differentiate each function individually and then add the results together. For example, if you are calculating the derivative of 𝑥² + 3𝑥, you would derive 𝑥² to get 2𝑥 and derive 3𝑥 to get 3, thus summing these gives you the final result of 2𝑥 + 3.
Examples & Analogies
Consider calculating costs for making a product where you have raw material costs and labor costs. If the material cost is represented by a function 𝑔(𝑥) and labor cost is represented by ℎ(𝑥), the total cost at any production level is the sum of these two costs. To find how costs change as production increases, you’d apply the Sum Rule to find the total cost change.
Product Rule
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Chapter Content
If 𝑓(𝑥) = 𝑔(𝑥)ℎ(𝑥), then
\[ \frac{d}{dx}[f(x)] = g'(x)h(x) + g(x)h'(x) \]
Example: 𝑑 [𝑥² ⋅ sin(𝑥)] = 2𝑥 sin(𝑥) + x² cos(𝑥).
Detailed Explanation
The Product Rule is used when differentiating a function that is the product of two other functions. According to this rule, the derivative is not simply the product of the derivatives; instead, you need to take the derivative of the first function, multiply it by the second function, and then add the product of the first function and the derivative of the second function. For example, if you differentiate 𝑥² ⋅ sin(𝑥), you first differentiate 𝑥² (which gives you 2𝑥), and multiply it by sin(𝑥), then add the product of 𝑥² and the derivative of sin(𝑥) (which is cos(𝑥)) yielding the complete derivative.
Examples & Analogies
Think of mixing two ingredients for a cake, say flour and sugar. The overall weight of the cake (function) can be considered a product of the weights of the individual ingredients. If you want to understand how changing the amount of flour or sugar affects the total weight of the cake, you'd use the Product Rule to calculate this change effectively.
Quotient Rule
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Chapter Content
If 𝑓(𝑥) = \frac{g(𝑥)}{ℎ(𝑥)} then
\[ \frac{d}{dx}[f(x)] = \frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2} \]
Example: \[ \frac{d}{dx}\left[ \frac{x^2}{x+1} \right] = \frac{2x(x+1) - x^2(1)}{(x+1)^2}. \]
Detailed Explanation
The Quotient Rule applies when differentiating functions that are in a quotient (division) form. It states that if 𝑓(𝑥) is the quotient of two functions 𝑔(𝑥) and ℎ(𝑥), the derivative is found by taking the derivative of the numerator, multiplying it by the denominator, subtracting the product of the numerator and the derivative of the denominator, all divided by the square of the denominator. This helps in maintaining clear relationships even when a function is divided by another. For example, with 𝑔(𝑥) = 𝑥² and ℎ(𝑥) = 𝑥 + 1, applying the quotient rule correctly yields the required derivative.
Examples & Analogies
Imagine you are analyzing speed as a ratio of distance to time. If distance varies with time, to find how fast speed changes overall when either distance (numerator) or time (denominator) changes, you would apply the Quotient Rule to calculate the changes effectively.
Chain Rule
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Chapter Content
If a function is composed of two or more functions, say 𝑓(𝑥) = 𝑔(ℎ(𝑥)), then
\[ \frac{d}{dx}[f(x)] = g'(h(x)) \cdot h'(x) \]
Example: \[ rac{d}{dx}[sin(x^2)] = cos(x^2) \cdot 2x. \]
Detailed Explanation
The Chain Rule is crucial for differentiating composite functions—functions made from one function nested inside another. To apply the Chain Rule, you differentiate the outer function first, then multiply it by the derivative of the inner function. For example, if you need to find the derivative of sin(𝑥²), you first differentiate sin(𝑓) to get cos(𝑓), and then you multiply that by the derivative of the inner function 𝑥² (which is 2𝑥). This properly captures how both layers of the function affect the rate of change.
Examples & Analogies
Think of a two-step recipe where you first mix ingredients (inner function) and then bake them (outer function). If you want to assess how changing an ingredient affects the final result, you'd need to know how each aspect influences the overall process—just like applying the Chain Rule helps elucidate layered functions.
Key Concepts
-
Power Rule: A fundamental rule for differentiating powers of x.
-
Sum Rule: The derivative of a sum is the sum of the derivatives.
-
Product Rule: A formula to differentiate products of functions.
-
Quotient Rule: A method for differentiating the division of two functions.
-
Chain Rule: A tool for differentiating composite functions.
Examples & Applications
Using the power rule, if f(x) = x^4, then f'(x) = 4x^3.
For f(x) = x^2 + 5, applying the sum rule gives f'(x) = 2x.
Memory Aids
Interactive tools to help you remember key concepts
Rhymes
To find the rate for x to the n, drop the n, count back down, then you can.
Stories
Imagine a climbing mountain. The higher you go, the steeper it feels—like finding the slope through calculus and using the power rule to navigate down!
Memory Tools
Silly Penguins Prefer Quick Cuddles (Sum, Power, Product, Quotient, Chain).
Acronyms
SPPQC - Sum, Power, Product, Quotient, Chain.
Flash Cards
Glossary
- Derivative
A measure of how a function changes as its input changes.
- Power Rule
A rule for differentiating functions of the form f(x) = x^n.
- Sum Rule
A rule that states the derivative of a sum of functions is the sum of their derivatives.
- Product Rule
A method for finding the derivative of the product of two functions.
- Quotient Rule
A method for finding the derivative of the quotient of two functions.
- Chain Rule
A technique for differentiating composite functions.
Reference links
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