The Chain Rule for Finding Derivatives | Chain Rule | Basic Calculus

Summary

This educational video provides a clear explanation of how to use the chain rule to find derivatives of composite functions. It walks through six examples, demonstrating the process of identifying the outer and inner functions, applying the chain rule, and simplifying the results. The examples range from basic to more complex functions, such as those involving powers and roots, offering viewers a comprehensive understanding of derivative calculations.

Takeaways

• 📚 The video is a tutorial on using the chain rule to find derivatives of composite functions.
• 🔍 The chain rule formula is given as the derivative of h(x) = g(f(x)) equals the derivative of g at f(x) times the derivative of f(x).
• 📝 The concept of 'inner' and 'outer' functions is introduced to help understand the composite function structure.
• 🔢 The first example demonstrates finding the derivative of a function involving a cubic term and a quadratic term inside a parenthesis.
• 📉 The second example shows the derivative of a function with a square root and a negative exponent, illustrating the power rule.
• 📈 The third example involves a function with a term raised to a fractional exponent, emphasizing the chain rule's application.
• 📌 The fourth example explains the derivative of a function with a square root, highlighting the use of the chain rule with exponents.
• 🔑 The fifth example covers the derivative of a cube root function, showing the simplification process using the chain rule.
• 📐 The sixth and final example in the script deals with the derivative of a function containing a square root of a quadratic expression.
• 👨‍🏫 The presenter, Prof D, encourages viewers to ask questions or seek clarifications in the comments section if needed.
• 👋 The video concludes with a sign-off from Prof D, indicating the end of the tutorial.

Q & A

What is the main topic of the video?

The main topic of the video is the application of the chain rule in calculus to find the derivative of composite functions.

What is the chain rule in derivatives?

The chain rule states that the derivative of a composite function h(x) = g(f(x)) is h'(x) = g'(f(x)) · f'(x), where g'(x) is the derivative of the outer function and f'(x) is the derivative of the inner function.

What is the first example given in the video to demonstrate the chain rule?

The first example is to find the derivative of f(x) = (5x^2 - 4x + 2)^3.

How is the derivative of the first example simplified using the chain rule?

The derivative is simplified by applying the chain rule to the outer function and the power rule to the inner function, resulting in 30x(5x^2 - 4x + 2)^2 - 12(5x^2 - 4x + 2).

What is the second example in the video, and how is the derivative found?

The second example is f(x) = (3x^2 - 4)^{1/2}. The derivative is found by applying the chain rule with the outer function being the square root and the inner function being 3x^2 - 4, resulting in 3x / √(3x^2 - 4).

What is the third example provided, and what is its derivative?

The third example is f(x) = 5x^4 - 7^{2/3}. The derivative is found by applying the chain rule to both terms, resulting in 40x^3 / 3(5x^4 - 7)^{1/3}.

In the fourth example, what is the function and its derivative?

The fourth example is f(x) = √(5x - 2). The derivative is 5 / 2√(5x - 2).

What is the function and its derivative in the fifth example?

The fifth example is f(x) = ∛(3x + 5). The derivative is 1 / 3(3x + 5)^{2/3}.

What is the sixth example in the video, and how is its derivative simplified?

The sixth example is f(x) = √(15x^2 - 6x + 2). The derivative is simplified by applying the chain rule and the power rule, resulting in (15x - 3) / 2√(15x^2 - 6x + 2).

What does the video suggest to do if viewers have questions or need clarifications?

The video suggests that viewers should leave their questions or requests for clarifications in the comment section below.

Outlines

00:00

📚 Introduction to Derivatives using the Chain Rule

This paragraph introduces the concept of derivatives and specifically the chain rule. The chain rule is a fundamental principle in calculus for finding the derivative of composite functions. The script explains that if a function h(x) is defined as g(f(x)), then the derivative of h(x) is the product of the derivative of g at f(x) and the derivative of f(x). The paragraph sets the stage for a series of examples that will demonstrate the application of the chain rule.

05:01

🔍 Derivative Calculation Examples with Chain Rule

This paragraph presents a series of examples to illustrate the application of the chain rule in calculating derivatives. The first example involves finding the derivative of a function composed of a cubic term and a linear term within a cubic function. The second example shows the derivative of a function with a square root and a linear term raised to a power. Each example is worked through step by step, applying the chain rule to find the derivative of the outer function and then multiplying by the derivative of the inner function. The explanations include the use of power rules and simplification of expressions.

10:03

📘 Advanced Derivative Examples and Simplification Techniques

The third paragraph continues with more complex examples of derivatives, including functions with exponents and roots. It demonstrates the process of applying the chain rule to functions involving powers and fractional exponents. The examples include the derivative of a function with a variable raised to a power minus a constant raised to a fractional power, and the derivative of a function with a square root of a quadratic expression. The paragraph emphasizes the importance of simplifying the result after applying the chain rule, showcasing the use of negative exponents and the final simplification of expressions.

Keywords

💡Derivative

A derivative in calculus represents the rate at which a function changes with respect to its variable. It is a fundamental concept in the study of functions and their behavior. In the video, the derivative is used to find the rate of change of composite functions, which is central to the application of the chain rule discussed.

💡Chain Rule

The chain rule is a fundamental theorem in calculus that allows the differentiation of composite functions. It states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function. The video script uses the chain rule to derive the derivatives of various functions, illustrating its application in multiple examples.

💡Composite Function

A composite function is created by applying one function to the result of another. For example, if h(x) = g(f(x)), then h is a composite of g and f. The video's theme revolves around finding the derivatives of such composite functions using the chain rule.

💡Outer Function

In a composite function, the outer function is the function that is applied last. In the context of the chain rule, it is the function whose derivative is taken first when differentiating the composite function. The script refers to the outer function when explaining the steps to find the derivative of a composite function.

💡Inner Function

The inner function is the function that is applied first within a composite function. It is differentiated after the outer function when applying the chain rule. The script uses the term 'inner function' to describe the part of the composite function that is differentiated after the outer function.

💡Power Rule

The power rule is a basic principle in calculus that allows for the differentiation of functions raised to a power. It states that the derivative of x^n, where n is a constant, is n*x^(n-1). The video script applies the power rule when differentiating functions that are raised to various powers.

💡Exponent

In mathematics, an exponent indicates the power to which a number is raised. In the context of the video, exponents are used to denote the power to which variables or constants are raised within functions, and understanding exponents is crucial for applying the power rule correctly.

💡Differentiation

Differentiation is the process of finding the derivative of a function. It is a key operation in calculus and is the focus of the video script, where the process of differentiating composite functions using the chain rule is demonstrated.

💡Simplifying

Simplification in mathematics refers to the process of making an expression easier to understand or work with, often by combining like terms or reducing fractions. In the script, simplification is mentioned as a step in the process of finding the derivative, where the resulting expressions are made simpler for clarity.

💡Binomial

A binomial is an algebraic expression that consists of two terms, typically involving a squared term and a linear term. In the video, binomials appear in the context of functions that are squared or raised to other powers, and the script demonstrates how to differentiate such expressions using the chain rule.

💡Contextualization

Contextualization is the process of placing a term or concept within a specific context to understand its meaning or significance better. In the video script, each keyword is contextualized within the theme of differentiation and the application of the chain rule, helping viewers to grasp the concepts more effectively.

Highlights

Introduction to the video on finding derivatives of functions using the chain rule.

Explanation of the chain rule formula for composite functions h(x) = g(f(x)).

Differentiating the outer function g(f(x)) and multiplying by the derivative of the inner function f(x).

Example 1: Finding the derivative of f(x) = (5x^2 - 4x + 2)^3 using the chain rule.

Applying the power rule to the outer function and the derivative of the inner function.

Simplifying the derivative by distributing and combining like terms.

Example 2: Derivative of f(x) = √(3x^2 - 4) using the chain rule.

Using the power rule for the square root and differentiating the inner function.

Simplifying the derivative by multiplying and combining terms.

Example 3: Derivative of f(x) = (5x^4 - 7)^(2/3) applying the chain rule.

Differentiating the outer function and inner function, then simplifying the result.

Example 4: Finding the derivative of f(x) = √(5x - 2) using the chain rule.

Applying the chain rule to the square root and differentiating the linear inner function.

Simplifying the derivative by multiplying by the reciprocal of the square root.

Example 5: Derivative of f(x) = ∛(3x + 5) using the chain rule.

Differentiating the cube root and the linear inner function, then simplifying.

Example 6: Derivative of f(x) = √(15x^2 - 6x + 2) applying the chain rule.

Using the power rule for the square root and differentiating the quadratic inner function.

Simplifying the derivative by distributing and combining terms for the quadratic function.

Conclusion of the video with an invitation for questions and clarifications in the comments.

Transcripts

[Music]

hello class welcome back to our channel

so for today's video

derivative

function by using

chain rule

okay so first

examinating formula

if h of x is equals to g of f of x

so on derivative down adding h of x is

equals

to the derivative of g of f of x

times the derivative of f of x

so if it's a b n

ma playing doughnut

chain rule but your given function

nothing i composite

okay

n

inner and outer function so in our

formula

jung g of x naught and young adding

outer function

while f of x atom and gain atting

inner function

so to get the derivative of a composite

function on gagavinylang guys is to get

the derivative out in outer function

times

so let's have our first example

find the derivative of f of x equals

the quantity of five x square minus four

x plus two

then cube

okay

so little guys american

you

uh five x square

minus four x plus two

you know

parenthesis guys

inner function

okay then formula

f of x

try nothing

so three times one your number not in

dito

that is positive three

so it will adding

five x square minus four x plus two then

after nothing ma play young acting power

rule magma minus one

satin exponent

outer function

the next afternoon

square that is 10x

then derivative adding negative 4x that

is negative 4

then derivative 2 is 0 so

a derivative

inner function

okay then simplifying that into guys

i you three for nothing you distribute

the starting binomial

so we have

five x square minus four x plus two

then three minus one that is positive

two

plus distribute not into three so we

have three times ten x that is thirty x

then three times negative four we have

negative twelve

tama

so eternal guys you adding

a final answer

see f prime of x

by using the chain rule

so let's proceed to example number two

we have f of x equals 3x square minus

four

raised to one half

so again applying the same procedure

first

derivative

uh outer function

so multiplying that in c one half dito

so that is one half

times three x square minus four

then see one half not n minus one

power rule then after nothing derivative

of a differentiating outer function

each chain nothing guys adding inner

function

okay multiply not n

derivative 3x square that is

a 6x

then derivative negative 4 that is 0

so

inner function 6x

then simplifying that into guys

we do nothing it multiplies one half k

six x

so that is

three x square minus four

then one half minus one

that is negative one half

then one half times six x so magic in

three x naught

okay so since multi-negative exponent

supply simplify not into guys we have

three x

over

the quantity of three x squared minus

four raised to one half

tama

negative exponent

so this will be our

final answer

now guys

so let's proceed to example number three

find the derivative of f of x equals 5 x

raised to 4 minus 7

raised to two thirds

okay

so applying the same procedure

uh get the derivative of the outer

function

so multiply not into theta guys so that

is two thirds

times five x raised to four minus seven

then two thirds minus one

okay

then each chain not in ionian derivative

no adding inner function

so that is derivative of 5x raised to 4

we have 20 x cubed

okay then derivative negative 7 is 0.

so at an ion in derivative

inner function

then simplifying that into guys

so putting nothing in multiplies totals

k20

so we have

5x raised to 4 minus 7

then two thirds minus one

that is negative one over three

comma

then two thirds times twenty we have

uh 40 over three

then x cube

okay

so little guys are made by a negative

exponent so if it's being put in an

elegant denominator

so simplify not into guys we have

a 40x cube

all over

we have 3 times the quantity of 5x

raised to 4

minus 7

raised to one third

okay so it in the guys

final answer

so next number four we have f of x

equals the square root of five x minus

two

so applying the same procedure first

chain rule error right

minus two

raised to one half

tama

so applying the chain rule so multiply

that into one half detail

so that is one half

times five x minus two

then your exponent not n magma minus 1

okay

inner function

derivative 5x minus 2 is

positive 5.

okay

so

nothing it multiplies one half k

positive five

so that is five x minus two

raised to one half minus one that is

negative one half

then one half times five that is

five over two

tama

then entire negative exponent

denominator so final answer nothing on

we have five over

so marathon to do times the quantity of

five x minus two

raised to one half

okay so with an iron guys you adding

f prime of x

listen then

next another example number five

we have f of x equals the cube root of

three x plus five

so applying the same procedure guys

first step

error right not in total exponential

form so that is

three x plus five

raised to one third

okay

so

nothing applies chain rule

so multiply not into so we have

one third

times three x plus five

then one third minus one

then

multiply nothing or each chain not any

derivative at the inner function so on

3 x plus 5

is equals to

positive 3

guys

multiply c 1 3 3 so putting my hands

so

we have

three x plus five

raised to one third minus one that is

negative two thirds

okay

so again by a negative exponent so for

simplifying

we have one over the quantity of three x

plus five

raised to positive

two thirds

okay so tonight

next number six

we have f of x equals the square root of

15 x square

minus six x plus two

so again

error right nothing you know adding a

square root to

exponent

so that is 15x squared minus 6x plus 2

raised to one half

power rule

in derivative the outer function

so that is one half

times fifteen x square minus six x plus

two

then one half minus one

okay

then get the derivative

inner function each chain not in n

so derivative 15 x square we have 30 x

then derivative of negative 6 x that is

negative 6

then derivative positive 2 is 0.

okay

so multiplying that you see one half did

something binomial

so we have 15 x square

minus six x plus two

raised to one half minus one that is

negative one half

comma then multiply that is one half

one half times thirty x that is fifteen

x then one half times negative six that

is

negative three

okay

so the two guys maritime negative

exponent

so

that is equals to 15 x minus 3

all over the quantity of 15 x square

minus 6 x plus 2

raised to one half

okay so ethernet guys you're

so this is the end of our video i hope

uh 19th and you guys

okay so if you have questions or

clarifications kindly put them in the

comment section below

so thank you guys for watching

this is prof d i'll catch you on the

flip side bye