How do you find the derivative of #f(x)=x^3+x^2# using the limit process?

1 Answer
Steve M
Nov 10, 2016

# f'(x )= 3x^2 + 2x #

Explanation:

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

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

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