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September 5th, 2017, 06:32 PM   #1
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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.
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Last edited by skipjack; September 5th, 2017 at 10:03 PM.
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September 5th, 2017, 11:37 PM   #2
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I don't really know what you are asking but ...

$(x+h)^n = \displaystyle \sum_{k=0}^n~\dbinom{n}{k}x^k h^{n-k}$

$(x+h)^n - x^n = \displaystyle \sum_{k=0}^{n-1}~\dbinom{n}{k}x^k h^{n-k}$

$\dfrac{(x+h)^n - x^n}{h} = \displaystyle \sum_{k=0}^{n-1}~\dbinom{n}{k}x^k h^{n-k-1}$

$\dfrac{d}{dx} x^n = \displaystyle \lim_{h\to 0} \left(\dfrac{(x+h)^n - x^n}{h}\right) =

\dbinom{n}{n-1}x^{n-1} = n x^{n-1}$
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