hw 14 - derivatives of logs section 2.9: problem 9 (1 point)\ndifferentiate $f(t)=\frac{10 + ln t}{10-ln…

Explanation:

Step1: Recall quotient - rule

The quotient - rule states that if $y=\frac{u}{v}$, then $y^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}$. Here, $u = 10+\ln t$, $v = 10-\ln t$.

Step2: Find $u^\prime$ and $v^\prime$

The derivative of $\ln t$ is $\frac{1}{t}$, and the derivative of a constant is 0. So, $u^\prime=\frac{1}{t}$ and $v^\prime=-\frac{1}{t}$.

Step3: Apply the quotient - rule

[ \begin{align*} f^\prime(t)&=\frac{\frac{1}{t}(10 - \ln t)-(10+\ln t)(-\frac{1}{t})}{(10-\ln t)^{2}}\ &=\frac{\frac{10}{t}-\frac{\ln t}{t}+\frac{10}{t}+\frac{\ln t}{t}}{(10 - \ln t)^{2}}\ &=\frac{\frac{20}{t}}{(10-\ln t)^{2}}\ &=\frac{20}{t(10 - \ln t)^{2}} \end{align*} ]

Answer:

$\frac{20}{t(10 - \ln t)^{2}}$