Differentiate the function. Select the correct answer.
$$f ( t ) = \frac { 2 + \ln t } { 5 - \ln t }$$
A) $f ^ { t } ( t ) = \frac { 7 } { t ( 1 - \ln t ) ^ { 2 } }$
B) $f ^ { t } ( t ) = \frac { 7 } { t ( 5 - \ln t ) ^ { 2 } }$
C) $f ^ { t } ( t ) = \frac { - 7 } { 5 t ( 2 - \ln t ) ^ { 3 } }$
D) $f ^ { t } ( t ) = \frac { - 3 } { t ( 5 - \ln t ) ^ { 2 } }$
E) $f ^ { t } ( t ) = \frac { 3 + 2 \ln t } { t ( 2 - \ln t ) ^ { 2 } }$

Question

Quizplus · Accepted Answer

We can use the quotient rule of differentiation. Let u = 2 + ln t and v = 5 - ln t. Then, we have:
$$f'(t) = \frac{v \frac{d}{dt}(u) - u \frac{d}{dt}(v)}{v^2} = \frac{(5 - \ln t)\left(\frac{1}{t}\right) - (2 + \ln t)\left(-\frac{1}{t}\right)}{(5 - \ln t)^2}$$
Simplifying this expression gives:
$$f'(t) = \frac{7}{t(5 - \ln t)^2}$$ 
Therefore, the correct answer is B.

Differentiate the Function f(t)=2+ln⁡t5−ln⁡tf ( t ) = \frac { 2 + \ln t } { 5 - \ln t }f(t)=5−lnt2+lnt​

Differentiate the Function $f ( t ) = \frac { 2 + \ln t } { 5 - \ln t }$