How do you use the definition of a derivative to find the derivative of #3x^2-5x+2#?

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Answer 1 · 2016-02-06T18:05:51

#6x-5#

The limit definition of a derivative states that the derivative of the function #f(x)# is

#f'(x)=lim_(hrarr0)(f(x+h)-f(x))/h#

Here, since #f(x)=3x^2-5x+2#, we see that #f(x+h)=3(x+h)^2-5(x+h)+2#.

Thus,

#f'(x)=lim_(hrarr0)(3(x+h)^2-5(x+h)+2-(3x^2-5x+2))/h#

Distribute terms in the numerator.

#f'(x)=lim_(hrarr0)(3(x^2+2xh+h^2)-5x-5h+2-3x^2+5x-2)/h#

#f'(x)=lim_(hrarr0)(3x^2+6xh+3h^2-5x-5h+2-3x^2+5x-2)/h#

Cancel terms.

#f'(x)=lim_(hrarr0)(6xh+3h^2-5h)/h#

Factor an #h#.

#f'(x)=lim_(hrarr0)(h(6x+3h-5))/h#

#f'(x)=lim_(hrarr0)6x+3h-5#

The limit can be be found by plugging in #0# for #h#.

#f'(x)=6x+3(0)-5#

#f'(x)=6x-5#