Question

How do you use the limit definition to find the derivative of `f(x)=sqrt(x+1)`?

Answer 1

`f'(x)=( 1 ) / ( 2sqrt(x+1) )`

Explanation:

By definition of the derivative `f'(x)=lim_(h rarr 0) ( f(x+h)-f(x) ) / h`
So with `f(x) = sqrt(x+1)` we have;

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

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