Question

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

Answer 1

`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) )`