Question

Use the first principle to find the derivative of the function f(x)=`20-:sqrt(x-1)`?

Answer 1

`f'(x)=-10/((x-1)^(3/2))`

Explanation:

The definition of a derivative (first principle)

`f'(x)=lim_(h->0)(f(x+h)-f(x))/h`

We want differentiate `f(x)=20/sqrt(x-1)`, therefore we seek

`f'(x)=lim_(h->0)20(1/sqrt(x+h-1)-1/sqrt(x-1))/h`

Let's rewrite the numerator

`NUM=1/sqrt(x+h-1)-1/sqrt(x-1)`

`=(sqrt(x-1)-sqrt(x+h-1))/(sqrt(x+h-1)sqrt(x-1))*(sqrt(x-1)+sqrt(x+h-1))/(sqrt(x-1)+sqrt(x+h-1))`

`=-h/(sqrt(x+h-1)sqrt(x-1)(sqrt(x-1)+sqrt(x+h-1))`

Thus

`f'(x)=lim_(h->0)20(-h/(sqrt(x+h-1)sqrt(x-1)(sqrt(x-1)+sqrt(x+h-1))))/h`

`=lim_(h->0)-(20)/(sqrt(x+h-1)sqrt(x-1)(sqrt(x-1)+sqrt(x+h-1)))`

`=-(20)/(sqrt(x-1)sqrt(x-1)(sqrt(x-1)+sqrt(x-1)))`

`=-(20)/(sqrt(x-1)sqrt(x-1)(2sqrt(x-1)))`

`=-20/(2(x-1)^(3/2))`

`=-10/((x-1)^(3/2))`