Question

10 POINTS!!! FULL ANSWER WITH FULL STEP BY STEP SOLUTION PLEASE

Answer 1

Let z = √(x^2 +2x +4).
Then your expression becomes
  y = (z^2)^z
  y = z^(2z)
And the derivative with respect to x is
  y' = z^(2z)*(2z' +2ln(z)z')
    = (2z')(1 +ln(z))(z^(2z))
Now
  z' = (2x +2)/(2√(x^2 +2x +4)
    = (x +1)/√(x^2 +2x +4)

So, your derivative is
  y' = 2(x +1)/√(x^2 +2x +4) * (1 + (1/2)ln(x^2 +2x +4)) * (x^2 +2x +4)^√(x^2 +2x +4)

For x = 0, this becomes
  2(1)/√4 * (1 +(1/2)ln(4)) * 4^√4
  = 16*(1 + ln(2))