How do you use the limit definition to find the derivative of #f(x)=sqrt(x-7)#?

1 Answer
Steve M
Nov 24, 2016

# f'(x)=( 1 ) / ( 2sqrt(x-7) )#

Explanation:

By definition of the derivative # f'(x)=lim_(h rarr 0) ( f(x+h)-f(x) ) / h #
So with # f(x) = sqrt(x-7) # we have;

# f'(x)=lim_(h rarr 0) ( sqrt((x+h)-7) - sqrt(x-7) ) / h #
# :. f'(x)=lim_(h rarr 0) ( sqrt(x+h-7) - sqrt(x-7) ) / h * ( sqrt(x+h-7) + sqrt(x-7) )/( sqrt(x+h-7) + sqrt(x-7) )#

# :. f'(x)=lim_(h rarr 0) ( (x+h-7) - (x-7) ) / (h * ( sqrt(x+h-7) + sqrt(x-7) ))#
# :. f'(x)=lim_(h rarr 0) ( h ) / (h * ( sqrt(x+h-7) + sqrt(x-7) ))#
# :. f'(x)=lim_(h rarr 0) ( 1 ) / ( sqrt(x+h-7) + sqrt(x-7) )#
# :. f'(x)=( 1 ) / ( sqrt(x-7) + sqrt(x-7) )#
# :. f'(x)=( 1 ) / ( 2sqrt(x-7) )#

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