Question

Please hjäp me to find the second derivative
f''(-1) to
f (x) = In (x ^ 2 +2 x +6)

Answer 1

f'(x) = 2(x+1)/(x^2+2x+6)
f''(x)=-2(x^2+2x-4)/(x^2+2x+6)^2
Therefore f''(-1)=2/5

Answer 2

f'(x)=(2x+2)/(x^2+2x+6)
now use the quotient rule to find second derivative
f''(x)={2(x^2+2x+6)-(2x+2)(2x+2)}/{(x^2+2x+6)^2}
={2x^2+4x+12-(4x^2+8x+4)}/{(x^2+2x+6)^2}
={-2x^2-4x+8}/{(x^2+2x+6)^2}

Answer 3

\[f(x) = \ln(x^{2} +2x+6) \Rightarrow f'(x)=\frac{1}{x^{2}+2x+6}(2x+2) = \frac{2x+2}{x^{2}+2x+6}\]
Now quotient rule:
\[f''(x) = \frac{(2)(x^{2}+2x+6)-(2x+2)(2x+2)}{(x^{2}+2x+6)^{2}}\]

Answer 4

Thanks, thanks and thanks