September 5th, 2017, 06:32 PM  #1 
Newbie Joined: Jun 2017 From: Perth, Australia Posts: 8 Thanks: 7  Gradient of a curve
Been given this investigation and I'm wondering what the in class extensions of this may be. What are some related concepts to this? I haven't attached the last page of the investigation, which is the main part of it, but I've attached a photo of an extension I thought of, which gets the rule for a cubic function, but I want to go further and understand more. Thanks.
Last edited by skipjack; September 5th, 2017 at 10:03 PM. 
September 5th, 2017, 11:37 PM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,382 Thanks: 1281 
I don't really know what you are asking but ... $(x+h)^n = \displaystyle \sum_{k=0}^n~\dbinom{n}{k}x^k h^{nk}$ $(x+h)^n  x^n = \displaystyle \sum_{k=0}^{n1}~\dbinom{n}{k}x^k h^{nk}$ $\dfrac{(x+h)^n  x^n}{h} = \displaystyle \sum_{k=0}^{n1}~\dbinom{n}{k}x^k h^{nk1}$ $\dfrac{d}{dx} x^n = \displaystyle \lim_{h\to 0} \left(\dfrac{(x+h)^n  x^n}{h}\right) = \dbinom{n}{n1}x^{n1} = n x^{n1}$ 

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curve, gradient 
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