How do I find the derivative of f(x) = 6x^2 - 1 using the definition of a derivative?

I understand that I am to plug-in components to make it look like f'(x) = 6(x+h)^2 - 1 - (6x^2 - 1)/h. But from this point onward I begin to get stuck. I am unsure as to what I am supposed to do, and the few places explaining what the next step is do not explain HOW to do that step.

1 Answer
Øko
May 28, 2018

#f'(x)=12x#

Explanation:

By definition of a derivative

#f'(x)=lim_(h->0)(f(x+h)-f(x))/h#

We want differentiate #color(blue)(f(x)=6x^2-1#, thus

#f'(x)=lim_(h->0)(6(x+h)^2-1-(6x^2-1))/h#

#color(white)(f'(x))=lim_(h->0)(6(x^2+h^2+2xh)-1-6x^2+1)/h#

#color(white)(f'(x))=lim_(h->0)(6x^2+6h^2+12xh-1-6x^2+1)/h#

#color(white)(f'(x))=lim_(h->0)(6h^2+12xh)/h#

#color(white)(f'(x))=lim_(h->0)(6h^2)/h+lim_(h->0)(12xh)/h#

#color(white)(f'(x))=lim_(h->0)6h+lim_(h->0)12x#

#color(white)(f'(x))=12x#

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