Calculus: Proving the Power Rule


You've probably started to study derivatives and you've come across this rule
which you were wondering how to proof.


 
So, how do we go about this?
If you remember the definition of what a derivative is (the rate at which y changes with respect to x), we may get to what we want to proof by using the binomial theorem [1][2] and differentiating .  

1º Step:

 

2º Step (using the binomial theorem [1] we expand the power):


Exercise: Using pascal's triangle and the binomial theorem try to complete the missing part for n=4

3º Step (factor out the , remove the and let approach 0):

I hope you have understood the proof and that you enjoyed the links below.
If you have any doubts, don't hesitate to contact me.


[1] Khan Academy Binomial Theorem 
[2] Binomial Coefficients (may help you with [1])

  ..

  xn 


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