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

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

1 Answer
Apr 27, 2018

-1/(sqrt(x^3))-1/x^21x31x2

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:(x+1)2:

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

We now have

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

Rewrite with negative exponents:

=1+2x^(-1/2)+x^-1=1+2x12+x1

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^-2ddx(x+1)2x=x32x2

=-1/(sqrt(x^3))-1/x^2=1x31x2