The Power Rule...

Okay this is Benofschool and going for another attempt to explain the Power Rule...
The following image shows the equation work that was used to derive the rule. I will explain each line after the image.

Line 1: Okay lets find the derivative of x to the power of n at x

Line 2: So we just throw it into the derivative formula like usual

Line 3: We have to expand the binomial in the brackets. There are 3 things to realize. First the first term will be x to the power of n regardless to what n is. Secondly, the next term will always have n as the coefficient and n-1 as the exponent and also multiplied by one h. Finally, every other term will have a unknown coefficient (this coefficient will become irrelevant and you will see later) and the exponent of h will increase by 1. So in the third term the power of h will be 2 and in the fourth it is 3 and so on

Line 4: Now we can get rid of x to the power of n because it is subtracted to zero by the other term.

Line 5: Now we factor out an h. Notice that the first term in Line 4 no long has an h variable in the term and every other term afterward still has an h variable remaining. This is the key to seeing the rule

Line 6: Now the h reduces and you are left with what was in the brackets in Line 5.

Line 7: Now we apply the limit and substitute all the h values with zero since there is no h in the brackets anymore. This will cause all terms containing the h variable to become zero. Since the first term does not contain an h variable, it won't be affected by the limit.

Line 8: This leaves us with the rule because all of the other terms are turned into zero thus not affecting the first term at all.

I hope this helped. If there is any more help required, please talk to Dr. Eviatar or myself for help...

Okay well I tried =P