Final steps Power Rule proof

[nosit]

Hello,

As per screenshot, I understand to arrive until this part:

x.k.x^(k-1) + x^(k).1

The problem is from this point below to:

k.x^k + x^k

For the second term is fine because the number is multiplying 1 so the result will be clearly x^k.

The problem is the first term from x.k.x^(k-1) to k.x^k

Could anyone please help me to understand it?

 

[JeffM]

Hello,

As per screenshot, I understand to arrive until this part:

x.k.x^(k-1) + x^(k).1

The problem is from this point below to:

k.x^k + x^k

For the second term is fine because the number is multiplying 1 so the result will be clearly x^k.

The problem is the first term from x.k.x^(k-1) to k.x^k

Could anyone please help me to understand it?

[MATH]x * k * x^{(k - 1)} \text { entails by commutativity of multiplication}[/MATH]
[MATH]k * x * x^{(k - 1)} \text { entails by laws of exponents}[/MATH]
[MATH]k * x^{\{1 + (k - 1)\}} \text { entails by commutativity of addition}[/MATH]
[MATH]k * x^{\{1 + (- 1 + k)\}} \text { entails by associativity of addition}[/MATH]
[MATH]k * x^{\{(1 - 1) + k\}} \text { entails by definition of additive inverse}[/MATH]
[MATH]k * x^{(0+k)} \text { entails by definition of additive identity (zero)}[/MATH]
[MATH]k * x^k = kx^k.[/MATH]
This is really basic algebra that should not have to be explained in such excruciating detail to a calculus student.

 

[nosit]

Hi @JeffM, indeed I have to improve a lot my (basic) algebra skills. I' ll keep studying.

Thank you for your help.

 

[Steven G]

@nosit, please do yourself a favor and study algebra before taking calculus. It will be worth skipping calculus for this semester in the long run.

 

[topsquark]

I agree. Most of my students had far more problems with algebra than they did with Calculus.

-Dan