Help with a couple natural log derivative problems !!!

[eromano33]

first, Find the derivative of f(t)= e^2ln(t + 1).

then, find the derivative of ln(radical all of the following)x^2 - 4

Finally, Simplify ln(2e^5x) + ln (e^3x /2)

Help if you could. I have the answers for each of these but I just don't know how to get there. Help is much appreciated- Final Exam on monday! Help!

 

[Unknown008]

Is the first one: \(\displaystyle f(t) = e^2 \ln(t+1)\)? e^2 is a constant in this one.

If you instead meant: \(\displaystyle f(t) = e^{2 \ln(t+1)}\), recall that if \(\displaystyle y = e^{f(x)}\), then \(\displaystyle y' = f'(x)e^{f(x)}\)


For the second one, is that: \(\displaystyle \ln \sqrt{x^2 - 4}\)? You can put it like this: \(\displaystyle \ln (x^2 - 4)^{\frac12} = \dfrac12 \ln (x^2-4)\)

Can you take it from here?


For the third one, when you have addition of log with the same base, you have an identity: \(\displaystyle \log_c a + \log_c b = \log_c (ab)\)