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The Chain Rule Chapter Notes | Quantitative Aptitude for Competitive Examinations - Bank Exams PDF Download
Welcome to Unit 3’s core concept: the Chain Rule! This essential technique helps us find derivatives of composite functions—functions nested within other functions. If you need a refresher on basic differentiation, revisit Unit 2 before diving in!
Understanding Composite Functions
Composite functions involve one function being applied to the result of another. Suppose we have two functions, f(x) and g(x). Their composite function, denoted (f ∘ g)(x), is defined as:
(f ∘ g)(x) = f(g(x))
Here, g(x) is the inner function, and f(x) is the outer function. This nesting is the key to applying the Chain Rule correctly.
Example of a Composite Function
Let’s look at an example with the functions:
- f(x) = x2
- g(x) = 3x + 1
For the composite function f(g(x)):
- The inner function g(x) = 3x + 1 takes an input x, multiplies it by 3, and adds 1.
- The outer function f(x) = x2 squares the result of g(x).
Thus, the composite function is:
f(g(x)) = (3x + 1)2
Grasping composite functions is crucial for mastering the Chain Rule. Let’s explore the rule itself!
What is the Chain Rule?
The Chain Rule is a differentiation technique for composite functions, similar to the Power Rule, Product Rule, or Quotient Rule from Unit 2. It allows us to differentiate functions where one function is embedded within another. The Chain Rule has two common notations:
Breaking this down:
- dy/dx is the derivative of the entire function with respect to x.
- u is the inner function.
- dy/du is the derivative of the outer function with respect to the inner function u.
- du/dx is the derivative of the inner function with respect to x.
Alternatively, in function notation:
This shows that we differentiate the outer function first (keeping the inner function intact), then multiply by the derivative of the inner function. If there are more nested functions, the process repeats.
Steps to Apply the Chain Rule
To differentiate using the Chain Rule, follow these steps:
- Identify the inner and outer functions.
- Differentiate the outer function with respect to the inner function.
- Differentiate the inner function with respect to x.
- Multiply the two derivatives to obtain the final result.
For a visual summary, check out this helpful diagram from Geeks for Geeks (image not included here; search online for Chain Rule visuals).
Question for Chapter Notes: The Chain RuleTry yourself: What does the Chain Rule help us find?View Solution
Practice Problems with the Chain Rule
Example 1:
Find the derivative of the following function with respect to x.
y = (x2+3x−1)2
- Define your inner and outer functions.
The inner function is u = x2 + 3x − 1. The outer function is y = u2. - Take the derivative of the outer function only!
dy/du = 2u = 2(x2+3x−1)
As you can see, we have not taken the derivative of the inner function u yet. - Take the derivative of the inner function with respect to x.
du/dx = 2x + 3 - Now multiply the two derivatives to get your final answer!
Example 2
Let f(x) = ex and g(x) = ln(x). Find the derivative of y = f(g(x)) with respect to x.
- Define your inner and outer functions.
In this problem, they are more explicit! f(x) is the outer function and g(x) is the inner function. - Take the derivative of the outer function.
f′(g(x)) = eg(x) = eln(x)
The derivative of e^x$ is just $e^x! - Take the derivative of the inner function with respect to x.
g′(x) = 1/x - Then multiply the two derivatives to get your final answer!
f′(g(x)⋅g′(x) = eln(x)⋅ 1/x
Example 3
Find the derivative of the following function; this one may be a bit tougher!
y = 4(5x3 + 2x2 + 6)2.
- Define your inner and outer functions.
The inner function is u = 5x3 + 2x2 + 6 and the outer function is y = 4u2. - Take the derivative of the outer function.
dy/du = 8u = 8(5x3 + 2x2 + 6) - Take the derivative of the inner function with respect to x.
du/dx = 15x2 + 4x - Now multiply the two derivatives to get your final answer!
Example 4
How about a function with a square root?
y = √(7x2).
- Define your inner and outer functions.
The inner function is u = 7x2 and the outer function is
- Take the derivative of the outer function.
- Take the derivative of the inner function with respect to x.
/fracdudx = 14x - Now multiply the two derivatives to get your final answer!
Example 5:
Try taking the derivative of the following function yourself!
f(x) = cos2(3x).
To help you out a little, the inner function is u = cos3x and the outer function is f(x) = u2.
Here’s the answer:
f′(x) = 2(cos3x)⋅ − 3sin(3x)
Question for Chapter Notes: The Chain RuleTry yourself: What is the inner function in the example y = (x² + 3x - 1)²?View Solution
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