Find the derivative using first principles? : #sqrt(x-5) #

1 Answer
Steve M
Nov 5, 2017

# d/dx sqrt(x-5) = 1/(2sqrt(x-5)) #

Explanation:

We seek:

# d/dx sqrt(x-5) #

By First Principles, using the limit definition:

# L = lim_(h rarr 0) (sqrt((x+h)-5) - sqrt(x-5))/h #

# \ \ = lim_(h rarr 0) (sqrt(x+h-5) - sqrt(x-5))/h *(sqrt(x+h-5) + sqrt(x-5))/(sqrt(x+h-5) + sqrt(x-5))#

# \ \ = lim_(h rarr 0) ((sqrt(x+h-5) - sqrt(x-5)) (sqrt(x+h-5) + sqrt(x-5))) / (h(sqrt(x+h-5) + sqrt(x-5)))#

# \ \ = lim_(h rarr 0) ( sqrt(x+h-5)^2 - sqrt(x-5)^2 ) / (h(sqrt(x+h-5) + sqrt(x-5)))#

# \ \ = lim_(h rarr 0) ( (x+h-5) - (x-5) ) / (h(sqrt(x+h-5) + sqrt(x-5)))#

# \ \ = lim_(h rarr 0) ( h ) / (h(sqrt(x+h-5) + sqrt(x-5)))#

# \ \ = lim_(h rarr 0) ( 1 ) / (sqrt(x+h-5) + sqrt(x-5))#

# \ \ = ( 1 ) / (sqrt(x+0-5) + sqrt(x-5))#

# \ \ = ( 1 ) / (sqrt(x-5) + sqrt(x-5))#

# \ \ = ( 1 ) / (2sqrt(x-5))#

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