How do you use the definition of a derivative to find the derivative of #sqrt(2x)#?

1 Answer
Steve M
Nov 19, 2016

# f'(x)=1 / ( sqrt(2x) )#

Explanation:

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

# f'(x)=lim_(h rarr 0) ( sqrt(2(x+h)) - sqrt(2x) ) / h #
# :. f'(x)=lim_(h rarr 0) ( sqrt(2x+2h) - sqrt(2x) ) / h #
# :. f'(x)=lim_(h rarr 0) ( sqrt(2x+2h) - sqrt(2x) ) / h * ( sqrt(2x+2h) + sqrt(2x) )/( sqrt(2x+2h) + sqrt(2x) )#
# :. f'(x)=lim_(h rarr 0) ( 2x+2h - 2x ) / (h( sqrt(2x+2h) + sqrt(2x) ))#
# :. f'(x)=lim_(h rarr 0) ( 2h ) / (h( sqrt(2x+2h) + sqrt(2x) ))#
# :. f'(x)=lim_(h rarr 0) 2 / ( sqrt(2x+2h) + sqrt(2x) )#
# :. f'(x)=2 / ( 2sqrt(2x) )#
# :. f'(x)=1 / ( sqrt(2x) )#

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