Question

Differentiate `sqrtx+1/sqrtx` without using first principles?

Answer 1

`(dy)/(dx)=(x-1)/(2xsqrtx)=(x-1)/(2x^(3/2)`

Explanation:

Here,

`(1)y=sqrtx+1/sqrtx=x^(1/2)+x^(-1/2)`

#=>(dy)/(dx)=1/2x^(1/2-1)-1/2x^(-1/2- 1)=1/2x^(-1/2)-1/2x^(-3/2)#

`=>(dy)/(dx)=1/(2x^(1/2))-1/(2x^(3/2))=(x-1)/(2x^(3/2)`

OR

`(2)y=sqrtx+1/sqrtx`

`=>y=(x+1)/sqrtx`

Using Quotient Rule:

`(dy)/(dx)=(sqrtx*d/(dx)(x+1)-(x+1)d/(dx)(sqrtx))/(sqrtx)^2`

`=>(dy)/(dx)=(sqrtx*1-(x+1)*1/(2sqrtx))/x`

`=>(dy)/(dx)=(2x-(x+1))/(2sqrtx*x)=(x-1)/(2xsqrtx`

`=>(dy)/(dx)=(x-1)/(2x^(3/2))`