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Using the limit definition, how do you differentiate $f(x)=x^2+3x+1$ ?

1 Answer

Apr 11, 2018

$f'(x)=2x+3$

Explanation:

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

In this case

$lim_(hrarr0)(f(x+h)-f(x))/h$

$=lim_(hrarr0)((x+h)^2+3(x+h)+1-x^2-3x-1)/h$

$=lim_(hrarr0)(x^2+2xh+h^2+3x+3h+1-x^2-3x-1)/h$

$=lim_(hrarr0)(2xh+h^2+3h)/h$

$=lim_(hrarr0)2x+h+3=2x+3$ .

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