How do you find f'(x) using the limit definition given # f(x) =sqrt (x+1)#?

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Answer 1 · 2016-07-04T14:58:43

#= 1/(2 sqrt (x+1) )#

#f(x) =sqrt (x+1)#

#f'(x) = lim_{h to 0} 1/h ( sqrt (x+ h+1) - sqrt (x+1))#

use the conjugate

#= lim_{h to 0} 1/h ( sqrt (x+ h+1) - sqrt (x+1)) *( sqrt (x+ h+1) + sqrt (x+1))/( sqrt (x+ h+1) + sqrt (x+1))#

#= lim_{h to 0} 1/h ( (x+ h+1) - (x+1)) /( sqrt (x+ h+1) + sqrt (x+1))#

#= lim_{h to 0} 1/h ( h) /( sqrt (x+ h+1) + sqrt (x+1))#

#= lim_{h to 0} 1/( sqrt (x+ h+1) + sqrt (x+1))#

#= 1/( sqrt (x+1) + sqrt (x+1))#

#= 1/(2 sqrt (x+1) )#