Question

Using the limit definition, how do you find the derivative of `f(x)=sqrt(x−3)`?

Answer 1

see explanation

Explanation:

f'(x) = `lim_(h→0)( f(x+h) -f(x))/h`

`=lim_(h→0) (sqrt(x+h-3) - sqrt(x-3))/h`

the aim here is to eliminate h from the denominator so that there is no division by zero as h→ 0.

consider multiplying numerator/denominator by

`sqrt(x+h-3) + sqrt(x-3)`
`color(black)("----------------------------------------------------------")`
numerator`= (sqrt(x+h-3) -sqrt(x-3))(sqrt(x+h-3) + sqrt(x-3))`
distribute brackets using FOIL

`= x+h-3 +sqrt((x-3))sqrt(x+h-3)) -sqrt(x-3)sqrt(x+h-3) -(x-3)`

`=x+h-3-x+3 =h`
`color(black)("-------------------------------------------------------")`
denominator

`sqrt(x-h+3) + sqrt(x-3)`
`color(black)("------------------------------------------------")`
now returning to f'(x)

f'(x) = `lim_(h→0) h/(h(sqrt(x-h+3) + sqrt(x-3))`
`= lim_(h→0) cancel(h)/(cancel(h)sqrt(x-h+3) + sqrt(x-3)`

`= 1/(sqrt(x-3) +sqrt(x-3)) = 1/(2sqrt(x-3))`