# Proof of Power Rule for 2 variables

• ak123456
In summary, the conversation discusses how to prove the equation u^n (x,y) = nu^(n-1) (x,y) u' (x,y) and whether there is a better method to do so. It is suggested to use the gradient (symbolized by \nabla) and the chain rule to prove the equation. Further steps in the proof are not discussed.
ak123456

## Homework Statement

u^n (x,y)=nu^(n-1) (x,y) u' (x,y)

## The Attempt at a Solution

can i set f (x,y)=u^n (x,y)
lnf =lnu^n
lnf=nlnU
f'/f=n/U
f'=fn/U -(U(^n) )(n/U)
after that i don't know how to continue and is there a better way to prove it

ak123456 said:

## Homework Statement

u^n (x,y)=nu^(n-1) (x,y) u' (x,y)
What do you mean by "u'(x,y)"? The gradient? The differential? "u' " is not defined for a function of two variables. If you mean the gradient, which is what I would think of as "the" derivative for a function of several variables, then yes, $\nabla u^n(x,y)= nu^{n-1}\nabla u$. That follows from the chain rule.

## The Attempt at a Solution

can i set f (x,y)=u^n (x,y)
lnf =lnu^n
lnf=nlnU
f'/f=n/U
f'=fn/U -(U(^n) )(n/U)
after that i don't know how to continue and is there a better way to prove it

## What is the Proof of Power Rule for 2 variables?

The Proof of Power Rule for 2 variables is a mathematical concept that states the derivative of a variable raised to a power is equal to the power multiplied by the variable raised to the power minus one.

## How is the Proof of Power Rule for 2 variables used in science?

In science, the Proof of Power Rule for 2 variables is used to find the rate of change of a function that involves two variables. This is particularly useful in fields such as physics and chemistry, where variables such as time and temperature may affect the outcome of an experiment.

## What is the formula for the Proof of Power Rule for 2 variables?

The formula for the Proof of Power Rule for 2 variables is d/dx (x^n) = nx^(n-1), where n represents the power or exponent and x represents the variable.

## Can the Proof of Power Rule for 2 variables be extended to more than 2 variables?

Yes, the Proof of Power Rule can be extended to any number of variables. The general formula for the Proof of Power Rule for n variables is d/dx (x^n) = nx^(n-1).

## What is the significance of the Proof of Power Rule for 2 variables?

The Proof of Power Rule for 2 variables is significant because it allows us to find the derivative of a function with two variables, which is useful in understanding the relationship between those variables and how they affect the outcome of the function. It is also a fundamental concept in calculus, which is widely used in various fields of science.

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