# Power Rule

\begin{aligned} \big({ x }^{ n }\big)\prime & = \lim _{ h\rightarrow 0 }{ \frac { { (x+h) }^{ n }-\big({ x }^{ n }\big) }{ h } } \\ \quad & = \lim _{ h\rightarrow 0 }{ \frac { \bigg(\displaystyle \sum _{ k=0 }^{ n }{ \binom{n}{k} { x }^{ n-k }{ h }^{ k } } \bigg)-{ x }^{ n } }{ h } } \\ \quad & = \lim _{ h\rightarrow 0 }{ \frac { { x }^{ n }+\bigg(\displaystyle \sum _{ k=1 }^{ n }{ \binom{n}{k} { x }^{ n-k }{ h }^{ k } } \bigg)-{ x }^{ n } }{ h } } \\ \quad & = \lim _{ h\rightarrow 0 }{ \frac \sum _{ k=1 }^{ n }{ h\binom{n}{k} { x }^{ n-k }{ h }^{ k-1 } } }{ h } } \\ \quad & = \lim _{ h\rightarrow 0 }{ \frac { h\bigg(\displaystyle \sum _{ k=1 }^{ n }{ \binom{n}{k} { x }^{ n-k }{ h }^{ k-1 } } \bigg) }{ h } } \\ \quad & = \lim _{ h\rightarrow 0 }{ \sum _{ k=1 }^{ n }{ \binom{n}{k} { x }^{ n-k }{ h }^{ k-1 } } } \\ \quad & = \lim _{ h\rightarrow 0 }{ \bigg(\binom{n}{1} { x }^{ n-1 }{ h }^{ 0 }+\binom{n}{2} { x }^{ n-2 }{ h }^{ 1 }+\binom{n}{3} { x }^{ n-3 }{ h }^{ 2 }+\cdots \bigg) } \\ \quad & = \lim _{ h\rightarrow 0 }{ \Bigg(n{ x }^{ n-1 }+h\bigg(\binom{n}{2} { x }^{ n-2 }+\binom{n}{3} { x }^{ n-3 }{ h }^{ 1 }+\cdots \bigg)\Bigg) } \\ \quad & = n{ x }^{ n-1 }+0\bigg(\binom{n}{2} { x }^{ n-2 }+\binom{n}{3} { x }^{ n-3 }{ 0 }^{ 1 }+\cdots \bigg) \\ \quad & = \boxed { n{ x }^{ n-1 } } \end{aligned Note by Gordon Chan
6 months, 4 weeks ago

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I see an error in the fifth line of your reasoning to the power rule. Since you are summing the h's over and over again you can simply factor it out so once it is factored out the h no longer appears in the summand.

- 5 months, 3 weeks ago

Oops. Thanks for pointing out.

- 5 months, 2 weeks ago

No Problem. I have one more question, do you like the Joy Of Problem solving course. I seem to love it. I want to know from you if there are other problem solving courses too on this. Thanks.

- 5 months, 2 weeks ago

I like it. I think you can try the 'Ace the AMC' course.

- 5 months, 2 weeks ago

The nervous system is responsible for sending, receiving and interpreting signals from all over the body and coordinates its action. It processes and intergrates information on a variety of levels, and directs the body to respond appropriately.

https://www.creative-bioarray.com/products/nervous-cells-list-148.htm?page=1

- 2 months, 1 week ago