DERIVATIVE OF RADICAL: THE CHAIN RULE
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
- ๐ 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).
- ๐ 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.
- โ Rewrite the given radical function y = (3x^2 + 5x - 4)^(1/3) in exponential form as y = (3x^2 + 5x - 4)^(1/3).
- ๐งฎ To find the derivative of y, use the chain rule: y' = (1/3) * (3x^2 + 5x - 4)^(-2/3) * (6x + 5).
- ๐ค Apply cross-multiplication to simplify the exponents when necessary, turning (1/3) - 1 into -2/3.
- ๐ The derivative of the first function is y' = (6x + 5) / (3(3x^2 + 5x - 4)^(2/3)) after simplification.
- ๐ 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.
- ๐ 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).
- ๐ To make the exponent positive, rewrite (2x^2 + 7)^(-1/4) as 1 / (2x^2 + 7)^(1/4).
- ๐ The final form of the derivative for the second function is y' = 3x / (2x^2 + 7)^(1/4).
- ๐ก Always simplify the derivative as much as possible for clarity and ease of use.
- ๐ 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
๐ 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'.
๐ 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
๐กDerivative
๐กChain Rule
๐กExponential Form
๐กCube Root
๐กSquare Root
๐กPower Function
๐กRadical
๐กDifferentiation
๐กComposite Function
๐กInner Function
๐กOuter Function
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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