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
okay so welcome to math storya and let's
have this topic
so finding the derivative of a radical
using the chain rule
then we have this even the cube root of
three x squared plus five x minus four
so first we need to rewrite this radical
in exponential form
so if we have this
square root of x in exponential form
so this one is x to the power
one half
so this is the numerator and this one is
the denominator of the exponent
so if we have this cube root of x
squared
so in exponential form this one is x to
the power
two over three
then four fourth root of two x plus one
so in exponential form so we have two x
plus one to the power
1 4
so therefore for this given
we can rewrite this one as y then equals
3x squared
plus 5x
minus 4 to the power 1 3.
then using the chain rule this formula
that y equals u to the power n
so to find the derivative that's n times
u to the power n minus 1 times the
derivative of u
so for this given
this is u
so u is equal to 3x squared
plus 5x minus 4
and for the derivative of u
so the derivative of 3x squared this one
is 6x
plus the derivative of 5x this one is 5
so therefore du is 6x plus 5
so to find out the derivative
using this formula and n
is the exponent
so we have y prime equals n
n is one third that's the exponent
then times u
u is three x squared
plus five x minus four
then we have n minus one
since n is one over three so we have now
one over three
minus one
so we're going to use cross
multiplication so over one
so we have one times one that's one then
minus three times one that's trading
over
three times one that's three so
therefore one third minus one is
negative
two over three
so this one is to the power negative two
over three
then times d u and d u
is six x plus five so we have six x
plus five
then simplify so we have y prime equals
so we need to multiply this one third
to this equation so we can write this
one as
six x
plus five
over three
then times
this three x squared
plus five x
minus four to the power negative two
over three
then if you want to make this exponent
positive so we just need to bring down
so we have y prime equals
the six x
plus five
then over
this three then times
three x squared
plus five x
minus four
to the power positive two over three
so this is now
the derivative
then for this given so to rewrite this
one in exponential form so we have y
then equals
two x squared plus seven then to the
power
three over four
then using the chain rule
so this is u
so u
is 2x squared plus 7 and du
so this one is
4x so du is equal to 4x
then to find the derivative using this
formula
so we have y prime equals
so n n is the exponent
that's 3 over 4 times u
two x squared plus seven
then n minus one
so we have three over four minus one so
that's over one
we have three
minus four this one is four so negative
one fourth so to the power negative
one fourth
then times d u and d u
is for
x then simplify so we have y prime
equals
so multiply this uh
3 over 4 times 4x so this one is
12x over 4 then times
2 x squared plus 7 to the power negative
1 port
then since we can simplify this 12 and 4
so y prime is equal to 3x
then times
2x squared plus 7 to the power negative
1 4
and to make this exponent positive so we
need to bring down
so y prime is equal to 3x then over
this 2x squared plus 7
to the power
1 4
so this is now the
derivative
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