Understanding of Power Rule

[nosit]

Hello guys,

I am now trying to understand the proof of Power Rule.

In the attachment I highlighted in yellow where I am stuck.

Why the (k+1).x^k becomes d/dx .x^k ?

Btw, as far as I know a derivative of a variable with respect to itself is 1

Thank you in advance.

 

[JeffM]

Suppose [MATH]y = x^k \text { and } z = x^{(k + 1)}.[/MATH]
[MATH]\text {But } \dfrac{dy}{dx} = kx^{(k-1)}.[/MATH]
[MATH]x^{(k+1)} = x * x^k \implies z = x * y \implies[/MATH]
[MATH]\dfrac{dz}{dx} = \dfrac{dx}{dx} * y + x * \dfrac{dy}{dx} \text { by Product Rule.}[/MATH]
[MATH]\therefore \dfrac{dz}{dx} = y + x * \dfrac{dy}{dx} = x^k + x * (kx^{(k-1)}) = \\ x^k + k(x^1 * x^{(k-1)}) = x^k + k(x^k) = (k + 1)x^k = (k + 1)x^{\{(k + 1) - 1\}}.[/MATH]

 

[nosit]

@JeffM thank you for your reply.

I stopped in the third line of your explanation. Why x^(k+1) = x.x^k ?

 

[JeffM]

@JeffM thank you for your reply.

I stopped in the third line of your explanation. Why x^(k+1) = x.x^k ?

Do you remember the rules of exponents from algebra?

[MATH]a^b * a^c = a^{(b+c)}.[/MATH]
[MATH]\therefore x^{(k+1)} = x^k * x^1 = x^1 * x^k = x^k.[/MATH]
This is one of those proofs that is so simple that you keep looking for something subtle in it. You say, "Is here where the subtlety lurks?" But there isn't anything subtle. It is just basic manipulation of exponents.

 

[pka]

I am now trying to understand the proof of Power Rule.
In the attachment I highlighted in yellow where I am stuck.
Why the (k+1).x^k becomes d/dx .x^k ?

It is not clear to me if you know the product rule\(^{**}\)? This depends upon that rule.
\( \begin{align*} \dfrac{d}{dx}\left[x\cdot x^k\right] &=\left(\dfrac{d}{dx}\right)x^k+x\left[\dfrac{d}{dx}x^k\right]** \\&=(1)x^k+k\left[x\cdot x^{k-1}\right]\\ &=x^k+k\cdot x^k\\&=(k+1)\cdot x^k\end{align*}\)

 

[Steven G]

[MATH]x^{k+1} = x*x^k then \dfrac{d}{dx}x^{k+1} = \dfrac{d}{dx}(x*x^k) = 1*x^k +x*kx^{k-1}[/MATH]

 

[nosit]

Do you remember the rules of exponents from algebra?

[MATH]a^b * a^c = a^{(b+c)}.[/MATH]
[MATH]\therefore x^{(k+1)} = x^k * x^1 = x^1 * x^k = x^k.[/MATH]
This is one of those proofs that is so simple that you keep looking for something subtle in it. You say, "Is here where the subtlety lurks?" But there isn't anything subtle. It is just basic manipulation of exponents.

Thank you for the explanation.

But why you say that x^1 . x^k = x^k ? The result according to the algebraic rule should be x^(1+k)

 

[Cubist]

Thank you for the explanation.

But why you say that x^1 . x^k = x^k ? The result according to the algebraic rule should be x^(1+k)

Look just above, @JeffM was responding to your question "Why x^(k+1) = x.x^k", so it seems like a simple mis-type or omission of an x from the right hand side. It should have been...

[MATH]x^{(k+1)} = x^k * x^1 = x^1 * x^k = x \cdot x^k[/MATH]

 

[nosit]

Look just above, @JeffM was responding to your question "Why x^(k+1) = x.x^k", so it seems like a simple mis-type or omission of an x from the right hand side. It should have been...

[MATH]x^{(k+1)} = x^k * x^1 = x^1 * x^k = x \cdot x^k[/MATH]

Oh, alright! Yeah, the explanation was good. I missed this x that multiplies x^k and just wanted to make sure it was a typo.

Thanks for the help.