more finding derivatives involving ln,logs

[Guest]

I need help on these:

v= log base 2 [( t^2+3t) ^1/2 ] <--this braacket part is NOT part of the log base, only the 2 is

v' 1/2 (t^2+ 3t) ^-1/2( 2t+3)/ [ t^2+3t)^1/2 (ln2) <--I dd this but am not sure if I'm correct, the answer is : (2t+3)/ [(2ln2(t^2+3t)]

and this question

2^x (logbase2) (x^4)

 

[soroban]

Hello, bittersweet!

\(\displaystyle v \:= \:\log_2(t^2\,+\,3t) ^{\frac{1}{2}}\)

Use one of the Log Properties to simplify it first . . .

\(\displaystyle \L v\;=\;\log_2(t^2\,+\,3t)^{\frac{1}{2}}\;=\;\frac{1}{2}\cdot\log_2(t^2\,+\,3t)\)

Then: \(\displaystyle \L\,v'\;=\;\frac{1}{2}\,\cdot\,\frac{1}{t^2\,+\,3t}\,\cdot\,\frac{1}{\ln2} \;= \;\frac{2t\,+\,3}{2\cdot\ln2\cdot(t^2\,+\,3t)}\)

\(\displaystyle f(x)\;=\;2^x\cdot\log_2(x^4)\)

Simplify first: \(\displaystyle \L\,f(x)\;=\;2^x\cdot4\cdot\log_2(x)\;=\;2^x\cdot2^2\cdot\log_2(x) \;= \;2^{x+2}\cdot\log_2(x)\)

Product Rule: \(\displaystyle \L\,f'(x)\;=\;2^{x+2}\cdot\frac{1}{x}\cdot\frac{1}{\ln2}\,+\,2^{x+2}\cdot\log_2(x) \;= \;2^{x+2}\left[\frac{1}{x\cdot\ln 2}\,+\,\log_2(x)\right]\)