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.
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
May 28, 2018
Explanation:
By definition of a derivative
#f'(x)=lim_(h->0)(f(x+h)-f(x))/h#
We want differentiate
#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#