Please help with this example to apply the laws of logarithms to simplify the function $\displaystyle f(x) =ln \sqrt\frac{9-x^2}{4+x^2}$. Then find it’s derivative.
I can't help you with the derivative, but here's the simply part:
you have: $\displaystyle \ln\sqrt{\frac{9-x^2}{4+x^2}}$
rewrite: $\displaystyle \ln\left(\left(\frac{9-x^2}{4+x^2}\right)^{\frac{1}{2}}\right)$
Thus: $\displaystyle \frac{1}{2}\ln\frac{9-x^2}{4+x^2}$
rewrite: $\displaystyle \frac{1}{2}\ln\frac{(3+x)(3-x)}{4+x^2}$
It is possible to go farther but I don't know how that might help...
Hello, Yogi_Bear_79!
Do you know the Laws of Logarithms?
Apply the laws of logarithms to simplify the function $\displaystyle f(x) = \ln\sqrt\frac{9-x^2}{4+x^2}$
Then find its derivative.
We have: .$\displaystyle f(x) \;= \;\ln\sqrt{\frac{9-x^2}{4+x^2}} \:=\:\ln\!\left(\frac{9-x^2}{4+x^2}\right)^{\frac{1}{2}} \;=\;\frac{1}{2}\cdot\ln\!\left(\frac{9-x^2}{4+x^2}\right)$
Then: .$\displaystyle f(x)\;=\;\frac{1}{2}\bigg[\ln(9-x^2) - \ln(4 + x^2)\bigg] $
Now differentiate it . . .