Math Help - Derivative

1. Derivative

Derivative of the function using logs

f(t)=e^2t_ln(t+1) and f(x)=ln(2x/x+1)

2. Hello factory1o1
Originally Posted by factory1o1
Derivative of the function using logs

f(t)=e^2t_ln(t+1) and f(x)=ln(2x/x+1)
I assume you mean $f(t) = e^{2t}\ln(t+1)$

in which case, use the product rule to get:

$f'(t)=e^{2t}\times\frac{1}{t+1}+ 2e^{2t}\ln(t+1)$

$=e^{2t}\Big(\frac{1}{t+1}+ 2\ln(t+1)\Big)$

To differentiate the second function, use the properties of logs first:

$f(x) = \ln\left(\frac{2x}{x+1}\right)$

$= \ln(2)+\ln(x) -\ln(x+1)$

$\Rightarrow f'(x) =\frac{1}{x}-\frac{1}{x+1}$

$=\frac{1}{x(x+1)}$