Question

what is the derivative using logarithmic differentiation of y= sqrt{x^2(x+1)(x+2)}, x>0

Answer 1

\[y=\sqrt{x^2(x+1)(x+2)}, x>0\]

Answer 2

first take the log of both sides and simplify...then differentiate

Answer 3

use the fact that \[\ln(A^n)=n\ln(A)\] and \[\ln(AB)=\ln(A)+\ln(B)\]

Answer 4

how would the right side look that is my problem i dont know where to put the 1/2 from the square root

for example would the right side be lnx+(1/2)ln(x+1)+(1/2)ln(x+2)

Answer 5

\[y=\sqrt{x^2(x+1)(x+2)}=(x^2(x+1)(x+2))^{1/2} \]

so \[\ln(y)=\ln\left[(x^2(x+1)(x+2))^{1/2} \right]\]

\[=\frac{1}{2}\ln\left[x^2(x+1)(x+2) \right]\]

\[=\frac{1}{2}\left[\ln(x^2)+\ln(x+1)+\ln(x+2) \right]\]

\[=\frac{1}{2}\left[2\ln(x)+\ln(x+1)+\ln(x+2) \right]\]


...yes :)

Answer 6

so you have

\[\ln(y)=\ln(x)+\frac{1}{2}\ln(x+1)+\frac{1}{2}\ln(x+2)\]

Answer 7

what did you get as a finale answer so i can compare to mine

Answer 8

what did you get and I will tell you if you are correct

Answer 9

dy/dx = \[(\sqrt{x^2(x+1)(x+2)})(\frac{ 1 }{ x }+\frac{ 1 }{ 2(x+1) }+\frac{ 1 }{ 2(x+2) })\]

Answer 10

yep

Answer 11

nice thanks again i might have another for you in a little