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[Miwaa]

what doe we call this rule?

y=x^ is y^(1)=nx^(n-1)

 

[🤖]

The rule you are referring to is known as the power rule or the derivative of a power rule. It is a fundamental rule in calculus that allows us to find the derivative of a function of the form y = x^n, where n is a constant.

In mathematical notation, the power rule states that if y = x^n, then the derivative of y with respect to x, denoted as dy/dx or y', is given by dy/dx = nx^(n-1).

This rule is derived using the chain rule, which is a general method for finding derivatives of composite functions. By applying the chain rule, we can differentiate each term of the function individually, resulting in the power rule formula.

The power rule is widely used in calculus to find derivatives of polynomial functions, as well as in various applications of mathematics and science involving rates of change.

 

[Miwaa]

The rule you are referring to is known as the power rule or the derivative of a power rule. It is a fundamental rule in calculus that allows us to find the derivative of a function of the form y = x^n, where n is a constant.

In mathematical notation, the power rule states that if y = x^n, then the derivative of y with respect to x, denoted as dy/dx or y', is given by dy/dx = nx^(n-1).

This rule is derived using the chain rule, which is a general method for finding derivatives of composite functions. By applying the chain rule, we can differentiate each term of the function individually, resulting in the power rule formula.

The power rule is widely used in calculus to find derivatives of polynomial functions, as well as in various applications of mathematics and science involving rates of change.

y=(x)=(x-9)/(\sqrt(x-3)) after refining it at point x=9

 

[🤖]

y=(x)=(x-9)/(\sqrt(x-3)) after refining it at point x=9

To refine the function y = (x-9)/sqrt(x-3) at the point x = 9, we need to find the value of the function at that specific point and determine its derivative at that point.

First, let's find the value of y at x = 9:

y = (x-9)/sqrt(x-3)
Substituting x = 9:
y = (9-9)/sqrt(9-3)
y = 0/3
y = 0

So, at x = 9, the value of y is 0.

Now, let's find the derivative of the function at x = 9. To do that, we can use the power rule and the chain rule.

Using the power rule, we differentiate the numerator and denominator separately:

Numerator:
d/dx (x-9) = 1

Denominator:
d/dx (sqrt(x-3)) = 1/(2sqrt(x-3))

Now, applying the chain rule to the entire function:

dy/dx = (1 * sqrt(x-3) - (x-9) * (1/(2sqrt(x-3)))) / (sqrt(x-3))^2

Simplifying further:

dy/dx = (sqrt(x-3) - (x-9) / (2sqrt(x-3))) / (x-3)

Now, let's find the derivative at x = 9:

dy/dx = (sqrt(9-3) - (9-9) / (2sqrt(9-3))) / (9-3)
dy/dx = (sqrt(6) - 0) / (2sqrt(6)) / 6
dy/dx = sqrt(6) / (2sqrt(6)) / 6
dy/dx = 1 / (2 * 6)
dy/dx = 1/12

Therefore, the refined function at x = 9 has a value of y = 0, and its derivative at x = 9 is dy/dx = 1/12.