242.2q.3 Find the derivative (1+ln{(t))/(1-ln{(t))

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In summary, the derivative of the given function is equal to 2 divided by t times the square of the difference between the natural logarithm of t and 1.
  • #1
karush
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$\tiny{242.2q.3}$
$\textsf{Find the derivative}\\$
\begin{align}
\displaystyle
y&=\frac{1+\ln{(t)}}{1-\ln{(t)}}
=-\frac{1+\ln(t)}{\ln(t)-1}=\frac{f}{g}\\
f&=1+\ln(t) \therefore f'=\frac{1}{t}\\
g&=\ln(t)-1 \therefore g'=\frac{1}{t}\\
y'&= \frac{f\cdot g' - f'\cdot g}{g^2}\\
&=\frac{(1+\ln(t))(1/t)-(\ln(t)-1)(1/t)}{(\ln(t)-1)^2} \\
&=\frac{1+\ln(t)-\ln(t)+1}{t(\ln(t)-1)^2}\\
&=\dfrac{2}{t\left(\ln\left(t\right)-1\right)^2}
\end{align}

$\textit{think this is ok, but suggestions before I cp it into overleaf?}$
 
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  • #2
The derivative is correct! (Yes)
 
  • #3
took me too long to do it...(Wasntme)
 

FAQ: 242.2q.3 Find the derivative (1+ln{(t))/(1-ln{(t))

1. What is the given function and what does it represent?

The given function is 242.2q.3 and it represents the derivative of the expression (1+ln{(t))/(1-ln{(t))}.

2. How do you find the derivative of this function?

To find the derivative of this function, we will use the quotient rule, which states that the derivative of a quotient of two functions is equal to the bottom function multiplied by the derivative of the top function minus the top function multiplied by the derivative of the bottom function, all divided by the bottom function squared.

3. What are the steps to find the derivative of this function?

The steps to find the derivative of this function are as follows:

  1. Identify the top and bottom functions.
  2. Apply the quotient rule: (bottom function)*(derivative of top function) - (top function)*(derivative of bottom function) / (bottom function)^2
  3. Simplify the resulting expression.

4. Can you simplify the resulting derivative?

Yes, the resulting derivative can be simplified by using algebraic manipulation and the properties of logarithms.

5. What is the final answer to the derivative of this function?

The final answer to the derivative of this function is (1/(t(1-ln{(t))^2)).

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