How do you find f'(x) using the definition of a derivative for #f(x)= 2x^2-x+5#?

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Answer 1 · 2015-10-17T23:20:03

#f'(x)=4x-1#

The definition of the derivative is

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

Proceeding as follows

#lim_(h->0)[[2(x+h)^2-(x+h)+5]-[2x^2-x+5]]/h#

#lim_(h->0)[[2(x^2+2xh+h^2)-x-h+5]-[2x^2-x+5]]/h#

#lim_(h->0)(2x^2+4xh+2h^2-x-h+5-2x^2+x-5)/h#

#lim_(h->0)(4xh+2h^2-h)/h#

#lim_(h->0)(h(4x+2h-1))/h#

#lim_(h->0)4x+2h-1=4x+2(0)-1=4x-1#