Question

Question on ln and Derivatives. (Hold on. Still typing the equations)

Answer 1

Why is the derivative of \[3^t\]
\[3^t \ln(3)\]

But the derivative of \[y = (\frac{ 4x-3 }{ 4x+5 })^(4x+1) \]
is
\[lny = (4x+1)\ln(\frac{ 4x-3 }{ 4x+5 })\]?

for the derivative of \[3^t\]
I thought it was tln(3) (because you bring down the exponent when you apply the ln), but you actually retain the original function. That doesn't make sense because in my second example, the derivative followed a different procedure (which illustrates the method that I used for my first example, but was told that it was wrong) and yet it is correct. I'm confused...

Answer 2

let me explain
let y=3^t
\[\ln y=\ln 3^t=t \ln 3\]
\[\frac{ 1 }{ y }\frac{ dy }{ dt }=1~\ln 3\]
\[\frac{ dy }{dt }=y \ln 3=3^t \ln 3\]

Answer 3

\[
3^t = (e^{\ln 3})^t = e^{t\ln 3}
\]You can differentiate this relatively simply with chain rule.

Consider \[
t^t = (e^{\ln t})^t = e^{t\ln t}
\]You have to use product rule here on top of the chain rule.