How do you find the derivative of #3x^2-5x+2# using the limit definition?

1 Answer
Steve M
Dec 3, 2016

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

Explanation:

By definition of the derivative # f'(x)=lim_(h rarr 0) ( f(x+h)-f(x) ) / h #
So with # f(x) = 3x^2 - 5x + 2 # we have;

# f'(x)=lim_(h rarr 0) ( {3(x+h)^2-5(x+h)+2 } - {3x^2 - 5x + 2 } ) / h #
# :. f'(x)=lim_(h rarr 0) ( {3(x^2+2hx+h^2)-5(x+h)+2 } - {3x^2 - 5x + 2 } ) / h #
# :. f'(x)=lim_(h rarr 0) ( {3x^2+6hx+3h^2-5x-5h+2 - 3x^2 + 5x - 2 } ) / h #

# :. f'(x)=lim_(h rarr 0) ( {6hx+3h^2-5h } ) / h #

# :. f'(x)=lim_(h rarr 0) ( 6x+3h-5 ) #

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

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