2 answers

How can i find the derivative of f(x)= x÷((x^(1/2) -1) ??

Question:

How can i find the derivative of f(x)= x÷((x^(1/2) -1) ??

Answers

See process below

Explanation:

#f(x)=x/(sqrtx-1)#

Apply cuotient derivative rule

#h/g=(h´·g-h·g´)/g^2# where #h=x# and #g=sqrtx-1#.

#f´(x)=(1·(sqrtx-1)-x·1/(2sqrtx))/(sqrtx-1)^2=(sqrtx-1-1/2sqrtx)/(x-2sqrtx+1)=(1/2sqrtx-1)/(x-2sqrtx+1)#

.

#1/(sqrt(x)-1)-sqrt(x)/(2(sqrt(x)-1)^2)#

Explanation:

#f(x)=x/(sqrt(x)-1)#

To find the derivertive of #f(x)# we need the quotient-rule

#f(x)=(k(x))/(g(x))#
#f'(x)=(g(x)k'(x)-k(x)g'(x))/(g(x))^2#
#k(x)=x;" "k'(x)=1#
#g(x)=sqrt(x)-1=x^(1/2)-1;" "g'(x)=1/2x^(-1/2)=1/(2sqrt(x))#

#f'(x)=((sqrt(x)-1) * (1)-(x) * (1/(2sqrt(x))))/(sqrt(x)-1)^2#
#=(sqrt(x)-1-x/(2sqrt(x)))/(sqrt(x)-1)^2=1/(sqrt(x)-1)-sqrt(x)/(2(sqrt(x)-1)^2)#

I'm sure you will do great:D
If you have any questions, feel free to ask:)

.

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