DERIVATIVE OF RADICAL: THE CHAIN RULE

MATHStorya
9 Mar 202206:03
EducationalLearning
32 Likes 10 Comments

TLDRThis video script from 'Math Story' dives into the process of finding the derivative of a radical function using the chain rule. The presenter starts by illustrating how to rewrite radicals in exponential form, such as the square root of x as x to the power of 1/2, and the cube root of x squared as x to the power of 2/3. The script then applies the chain rule formula, y = u^n, to find the derivative of y with respect to x. Two examples are given: one with a cube root and another with a fourth root. The presenter carefully explains how to calculate the derivative by multiplying the exponent (n) with the function inside the radical (u), then adjusting for the power (n-1), and finally multiplying by the derivative of u (du/dx). The process is detailed, with the presenter showing each step of the simplification, including changing negative exponents to positive by taking the reciprocal. The script concludes with the final expressions for the derivatives of the given functions, providing a clear understanding of the derivative calculation for radicals.

Takeaways
  • ๐Ÿ“š Convert radicals to exponential form to apply the chain rule for derivatives. For example, the square root of x is written as x^(1/2).
  • ๐Ÿ”‘ The chain rule formula is given by n * u^(n-1) * du/dx, where n is the exponent, u is the function inside the radical, and du/dx is the derivative of u.
  • โœ… Rewrite the given radical function y = (3x^2 + 5x - 4)^(1/3) in exponential form as y = (3x^2 + 5x - 4)^(1/3).
  • ๐Ÿงฎ To find the derivative of y, use the chain rule: y' = (1/3) * (3x^2 + 5x - 4)^(-2/3) * (6x + 5).
  • ๐Ÿค” Apply cross-multiplication to simplify the exponents when necessary, turning (1/3) - 1 into -2/3.
  • ๐Ÿ“ˆ The derivative of the first function is y' = (6x + 5) / (3(3x^2 + 5x - 4)^(2/3)) after simplification.
  • ๐ŸŒŸ For the second function y = (2x^2 + 7)^(3/4), rewrite it using the chain rule with u = 2x^2 + 7 and du/dx = 4x.
  • ๐Ÿ“ The derivative of y for the second function is y' = (3/4) * 4x * (2x^2 + 7)^(-1/4), which simplifies to 3x * (2x^2 + 7)^(-1/4).
  • ๐Ÿ“‰ To make the exponent positive, rewrite (2x^2 + 7)^(-1/4) as 1 / (2x^2 + 7)^(1/4).
  • ๐Ÿ“ The final form of the derivative for the second function is y' = 3x / (2x^2 + 7)^(1/4).
  • ๐Ÿ’ก Always simplify the derivative as much as possible for clarity and ease of use.
  • ๐Ÿ“– Understanding the chain rule and how to apply it to radicals is crucial for calculus and mathematical problem-solving.
Q & A
  • What is the main topic of the video script?

    -The main topic of the video script is finding the derivative of a radical using the chain rule.

  • How is the cube root of x represented in exponential form?

    -The cube root of x is represented in exponential form as x to the power of two over three (x^(2/3)).

  • What is the general formula for the derivative using the chain rule?

    -The general formula for the derivative using the chain rule is n times u to the power n minus 1 times the derivative of u, where y = u^n.

  • What is the exponential form of the expression 3x squared plus 5x minus 4 to the power of 1/3?

    -The exponential form of the expression is (3x^2 + 5x - 4)^(1/3).

  • How is the derivative of u (where u = 3x^2 + 5x - 4) calculated?

    -The derivative of u, denoted as du, is calculated as the derivative of 3x^2 (which is 6x) plus the derivative of 5x (which is 5), so du = 6x + 5.

  • What is the final simplified form of the derivative of y with respect to x for the first example given in the script?

    -The final simplified form of the derivative, denoted as y', is (6x + 5) / ((3x^2 + 5x - 4)^(2/3)).

  • How is the exponent made positive when expressing the derivative?

    -The exponent is made positive by using the property that a negative exponent is equivalent to taking the reciprocal of the base raised to the positive exponent.

  • What is the exponential form of the expression 2x squared plus 7 to the power of 3/4?

    -The exponential form of the expression is (2x^2 + 7)^(3/4).

  • What is the derivative of u (where u = 2x^2 + 7)?

    -The derivative of u, denoted as du, is the derivative of 2x^2 (which is 4x), so du = 4x.

  • What is the final simplified form of the derivative of y with respect to x for the second example given in the script?

    -The final simplified form of the derivative, denoted as y', is (3x * (2x^2 + 7)^(-1/4)) or (3x / ((2x^2 + 7)^(1/4))).

  • What is the significance of using the chain rule in finding derivatives of radicals?

    -The chain rule is significant as it allows us to find the derivative of composite functions, especially when dealing with radicals or fractional exponents, by breaking down the function into simpler components.

  • How does the script demonstrate the process of differentiating an expression with a radical?

    -The script demonstrates the process by first rewriting the radical expression in exponential form, identifying u and du, applying the chain rule formula, and then simplifying the expression to find the derivative y'.

Outlines
00:00
๐Ÿ“š Finding Derivatives Using the Chain Rule

This paragraph introduces the concept of finding derivatives of radicals using the chain rule. It begins with the expression 'the cube root of three x squared plus five x minus four' and demonstrates how to rewrite it in exponential form as 'y equals 3x squared plus 5x minus 4 to the power of 1/3'. The chain rule formula 'y equals u to the power n' is then applied, where 'u' is the expression inside the radical, and 'n' is the exponent. The derivative 'du' is calculated by differentiating 'u', resulting in '6x plus 5'. The final derivative 'y prime' is found by multiplying the exponent 'n' (one third) with 'u' and adjusting for the power of 'n minus one', which simplifies to 'negative two over three'. The derivative is then simplified to 'six x plus five over three times the given expression to the power of negative two over three'.

05:02
๐Ÿ” Derivative of a Radical Expression with Exponential Form

The second paragraph continues the theme of derivatives but with a different radical expression, 'two x squared plus seven to the power of three over four'. It starts by expressing 'u' as '2x squared plus 7' and its derivative 'du' as '4x'. Applying the chain rule again, the derivative 'y prime' is calculated by multiplying the exponent '3/4' with 'u' and adjusting for 'n minus one', which results in 'negative one fourth'. The final expression for the derivative 'y prime' is simplified to '3x times 2x squared plus 7 to the power of negative one fourth'. To make the exponent positive, the expression is manipulated algebraically to '3x over 2x squared plus 7 to the power of one fourth', providing the final form of the derivative.

Mindmap
Interplay between exponents and roots in calculus
Role of polynomials in calculus
Simplifying expressions with exponents
Changing negative exponents to positive
Technique for simplifying fractional exponents
Simplifying the resulting derivative
Using the chain rule for the derivative
Rewriting the expression in exponential form
Expression: Fourth root of (2x^2 + 7)
Simplifying the derivative expression
Application of the chain rule to find the derivative
Rewriting the expression in exponential form
Expression: Cube root of (3x^2 + 5x - 4)
Use of exponents to simplify derivative calculations
Rewriting radicals in exponential form
Application of the chain rule to find derivatives
Definition of the chain rule in calculus
Derivatives as a measure of change
Importance of understanding derivatives in calculus
Exponents and Roots
Understanding of Polynomials
Exponent Manipulation
Cross Multiplication
Second Example: Fourth Root of a Polynomial
First Example: Cube Root of a Polynomial
Exponential Form
Chain Rule
Introduction to Derivatives
Mathematical Concepts
Derivative Simplification
Example Calculations
Derivative of Radicals
Mathematical Derivatives
Alert
Keywords
๐Ÿ’กDerivative
A derivative in calculus represents the rate at which a function changes with respect to a variable. It is a fundamental concept used to analyze the behavior of functions and is central to the video's theme of differentiating radical expressions. In the script, the process of finding the derivative of various functions, such as radicals and polynomials, is explained in detail.
๐Ÿ’กChain Rule
The chain rule is a fundamental theorem in calculus for differentiating 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 script uses the chain rule to find derivatives of functions that are expressed as roots or powers, which is a key technique demonstrated in the video.
๐Ÿ’กExponential Form
Exponential form is a way of expressing numbers in terms of a base raised to an exponent. In the context of the video, it is used to rewrite roots and powers of variables in a form that makes differentiation easier. For example, the cube root of x is rewritten as x to the power of 2/3, which is then used to apply the chain rule for differentiation.
๐Ÿ’กCube Root
A cube root is a mathematical operation that finds a number which, when multiplied by itself three times, gives the original number. In the video, the cube root of an expression is used as an example to illustrate the process of finding derivatives using the chain rule. It is an important part of the script's exploration of radical functions.
๐Ÿ’กSquare Root
The square root of a number is a value that, when multiplied by itself, gives the original number. It is a specific case of a more general concept of roots. In the script, the square root of x is expressed in exponential form as x to the power of 1/2, which is then used to demonstrate the differentiation process.
๐Ÿ’กPower Function
A power function is a function of the form f(x) = x^n, where n is a rational number. The concept is essential in calculus for understanding how to differentiate functions that involve raising a variable to a power. The video script discusses power functions in the context of rewriting radicals in exponential form and differentiating them using the chain rule.
๐Ÿ’กRadical
A radical, in mathematics, refers to the root of a number. It is often denoted by a symbol (โˆš) for square roots or with other notations for cube roots and higher order roots. The video focuses on finding the derivatives of radical expressions, which involves converting them into exponential form and applying differentiation rules.
๐Ÿ’กDifferentiation
Differentiation is the process of finding the derivative of a function, which gives the rate at which the function is changing at a given point. It is a key concept in calculus and is the main focus of the video. The script provides a step-by-step guide on how to differentiate various types of functions, including radicals and power functions.
๐Ÿ’กComposite Function
A composite function is a function composed of two or more functions, where the output of one function becomes the input of the next. The concept is important in the video because the chain rule is used to differentiate composite functions. The script demonstrates the application of the chain rule to find derivatives of such functions.
๐Ÿ’กInner Function
In the context of composite functions, the inner function is the function that is applied first, and its output is then used as the input for the outer function. The script discusses the concept of the inner function in relation to the chain rule, where the derivative of the inner function is a key component in finding the derivative of the composite function.
๐Ÿ’กOuter Function
The outer function is the function that is applied to the result of the inner function in a composite function. The video uses the concept of the outer function to explain how the chain rule operates. Specifically, the derivative of the outer function is multiplied by the derivative of the inner function to find the overall derivative.
Highlights

Introduction to finding the derivative of a radical using the chain rule.

Rewriting the radical expression in exponential form for better understanding.

Explanation of converting square root and cube root to exponential form.

Derivation of the general formula for the derivative using the chain rule: n * u^(n-1) * du/dx.

Identification of 'u' as the inner function in the given radical expression.

Calculation of the derivative of 'u' (du/dx) for the given expression.

Application of the chain rule formula to find the derivative of the given radical expression.

Use of cross-multiplication to handle fractional exponents in the derivative.

Simplification of the derivative expression to its final form.

Rewriting the given radical expression in exponential form for a different example.

Derivation of the inner function 'u' and its derivative for the second example.

Application of the chain rule to the second example to find the derivative.

Adjustment of the exponent to a positive value for clarity in the derivative expression.

Final simplification of the derivative for the second example.

Emphasis on the importance of simplifying expressions and correctly applying the chain rule.

The process demonstrates a clear method for differentiating radicals using the chain rule.

The transcript provides a step-by-step guide that is easy to follow for understanding derivatives of radicals.

The use of two different examples helps to solidify the understanding of the chain rule application.

Transcripts
00:00

okay so welcome to math storya and let's

00:03

have this topic

00:04

so finding the derivative of a radical

00:07

using the chain rule

00:09

then we have this even the cube root of

00:11

three x squared plus five x minus four

00:15

so first we need to rewrite this radical

00:18

in exponential form

00:20

so if we have this

00:22

square root of x in exponential form

00:26

so this one is x to the power

00:29

one half

00:30

so this is the numerator and this one is

00:33

the denominator of the exponent

00:36

so if we have this cube root of x

00:39

squared

00:40

so in exponential form this one is x to

00:44

the power

00:45

two over three

00:47

then four fourth root of two x plus one

00:51

so in exponential form so we have two x

00:54

plus one to the power

00:57

1 4

00:58

so therefore for this given

01:01

we can rewrite this one as y then equals

01:05

3x squared

01:07

plus 5x

01:09

minus 4 to the power 1 3.

01:13

then using the chain rule this formula

01:16

that y equals u to the power n

01:19

so to find the derivative that's n times

01:22

u to the power n minus 1 times the

01:24

derivative of u

01:26

so for this given

01:28

this is u

01:31

so u is equal to 3x squared

01:34

plus 5x minus 4

01:38

and for the derivative of u

01:41

so the derivative of 3x squared this one

01:44

is 6x

01:46

plus the derivative of 5x this one is 5

01:50

so therefore du is 6x plus 5

01:54

so to find out the derivative

01:57

using this formula and n

02:00

is the exponent

02:02

so we have y prime equals n

02:06

n is one third that's the exponent

02:10

then times u

02:12

u is three x squared

02:15

plus five x minus four

02:18

then we have n minus one

02:21

since n is one over three so we have now

02:24

one over three

02:26

minus one

02:27

so we're going to use cross

02:29

multiplication so over one

02:33

so we have one times one that's one then

02:35

minus three times one that's trading

02:37

over

02:39

three times one that's three so

02:40

therefore one third minus one is

02:42

negative

02:44

two over three

02:46

so this one is to the power negative two

02:49

over three

02:51

then times d u and d u

02:54

is six x plus five so we have six x

02:58

plus five

03:00

then simplify so we have y prime equals

03:05

so we need to multiply this one third

03:08

to this equation so we can write this

03:11

one as

03:13

six x

03:14

plus five

03:16

over three

03:17

then times

03:19

this three x squared

03:22

plus five x

03:24

minus four to the power negative two

03:26

over three

03:28

then if you want to make this exponent

03:30

positive so we just need to bring down

03:34

so we have y prime equals

03:37

the six x

03:39

plus five

03:40

then over

03:42

this three then times

03:45

three x squared

03:47

plus five x

03:49

minus four

03:50

to the power positive two over three

03:54

so this is now

03:56

the derivative

03:59

then for this given so to rewrite this

04:01

one in exponential form so we have y

04:04

then equals

04:06

two x squared plus seven then to the

04:08

power

04:10

three over four

04:14

then using the chain rule

04:16

so this is u

04:18

so u

04:19

is 2x squared plus 7 and du

04:23

so this one is

04:25

4x so du is equal to 4x

04:29

then to find the derivative using this

04:31

formula

04:33

so we have y prime equals

04:36

so n n is the exponent

04:39

that's 3 over 4 times u

04:42

two x squared plus seven

04:45

then n minus one

04:47

so we have three over four minus one so

04:50

that's over one

04:52

we have three

04:54

minus four this one is four so negative

04:58

one fourth so to the power negative

05:02

one fourth

05:03

then times d u and d u

05:07

is for

05:08

x then simplify so we have y prime

05:12

equals

05:14

so multiply this uh

05:16

3 over 4 times 4x so this one is

05:20

12x over 4 then times

05:23

2 x squared plus 7 to the power negative

05:26

1 port

05:27

then since we can simplify this 12 and 4

05:31

so y prime is equal to 3x

05:35

then times

05:37

2x squared plus 7 to the power negative

05:40

1 4

05:42

and to make this exponent positive so we

05:44

need to bring down

05:46

so y prime is equal to 3x then over

05:51

this 2x squared plus 7

05:55

to the power

05:56

1 4

05:57

so this is now the

05:59

derivative