DERIVATIVE OF RADICAL: THE CHAIN RULE

MATHStorya
9 Mar 202206:03
EducationalLearning
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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
  • 📚 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).
  • 🔑 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.
  • ✅ Rewrite the given radical function y = (3x^2 + 5x - 4)^(1/3) in exponential form as y = (3x^2 + 5x - 4)^(1/3).
  • 🧮 To find the derivative of y, use the chain rule: y' = (1/3) * (3x^2 + 5x - 4)^(-2/3) * (6x + 5).
  • 🤔 Apply cross-multiplication to simplify the exponents when necessary, turning (1/3) - 1 into -2/3.
  • 📈 The derivative of the first function is y' = (6x + 5) / (3(3x^2 + 5x - 4)^(2/3)) after simplification.
  • 🌟 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.
  • 📝 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).
  • 📉 To make the exponent positive, rewrite (2x^2 + 7)^(-1/4) as 1 / (2x^2 + 7)^(1/4).
  • 📐 The final form of the derivative for the second function is y' = 3x / (2x^2 + 7)^(1/4).
  • 💡 Always simplify the derivative as much as possible for clarity and ease of use.
  • 📖 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
📚 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
🔍 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
Keywords
💡Derivative
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.
💡Chain 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.
💡Exponential 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.
💡Cube 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.
💡Square 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.
💡Power 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.
💡Radical
A radical, in mathematics, refers to the root of a number. It is often denoted by a symbol (√) 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.
💡Differentiation
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.
💡Composite 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.
💡Inner 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.
💡Outer 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

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