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

Question

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

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Answer 1 · 2018-04-27T17:17:32

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

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#