Use the first principle to find the derivative of the function f(x)= $20 \div \sqrt{x - 1}$ $20-:sqrt(x-1)$ ?

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Answer 1 · 2018-02-28T08:54:21

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

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

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

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

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

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

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

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

Use the first principle to find the derivative of the function f(x)=20-:sqrt(x-1)20÷√x−120-:sqrt(x-1)?