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 $h$ .

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

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

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