Question

Differentiate `(sqrt(x) + 1 )^2/ x` with respect to `X.`?

Differentiate `(sqrt(x) + 1 )^2/ x` with respect to `X.`

Answer 1

`-1/(sqrt(x^3))-1/x^2`

Explanation:

We can avoid use of the Quotient Rule and Chain rule rather easily in this problem, it avoids a lot of unnecessary work:

Expand `(sqrtx+1)^2:`

`(sqrtx+1)(sqrtx+1)=x+2sqrtx+1`

We now have

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

Rewrite with negative exponents:

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

Differentiate, noting how we now only need to apply the Power Rule and the fact that the derivative of a constant is zero:

`d/dx(sqrtx+1)^2/x=-x^(-3/2)-x^-2`

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