Use the first principle to find the derivative of the function f(x)=#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))#