How do you use the limit definition of the derivative to find the derivative of #f(x)=-3x^2+x+5#?

1 Answer
Steve M
Nov 21, 2016

# f'(x)=-6x+1 #

Explanation:

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

# f'(x)=lim_(h rarr 0) ( {-3(x+h)^2+(x+h)+5 } - {-3x^2+x+5 } ) / h #
# :. f'(x)=lim_(h rarr 0) ( {-3(x^2+2hx+h^2)+x+h+5 } - {-3x^2+x+5 } ) / h #
# :. f'(x)=lim_(h rarr 0) ( -3x^2-6hx-3h^2+x+h+5 +3x^2-x-5 ) / h #
# :. f'(x)=lim_(h rarr 0) ( -6hx-2h^2+h ) / h #
# :. f'(x)=lim_(h rarr 0) ( -6x-2h+1 ) #
# :. f'(x)=-6x+1 #

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