How do you find f'(x) using the limit definition given #f (x) = -(2/3x) #?

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Answer 1 · 2016-06-21T10:46:01

# - 2/3 #

the classic definition is #f'(x) = (f(x+h) - f(x) )/ h |_{h \to 0}#

here that becomes

#f'(x) =( - (2/3 (x + h)) -( - 2/3 x) )/ h |_{h \to 0}#

# = (- (2/3 h) )/ h |_{h \to 0}#

# = - 2/3 #

which is what you would expect :-)

crucially because we are in the limit # h \to 0 # but # h \ne 0 # we can do that division and cancel the h's.