Question

attempt 1: 10 attempts remaining. the productivity score $p(t)$ of employees in a tech company depends on the number of hours $t$ of skill training provided annually. this relationship is modeled by the function: $p(t)=88 + 19ln(t)$ where $t$ is the number of hours of training and $t>0$. a) find the derivative $p(t)$. $p(t)=$ b) compute $p(5)$. round to two decimal places as necessary. $p(5)=$

Explanation:

Step1: Recall derivative rules

The derivative of a constant $C$ is $0$, i.e., $\frac{d}{dt}(C)=0$, and the derivative of $\ln(t)$ is $\frac{1}{t}$, i.e., $\frac{d}{dt}(\ln(t))=\frac{1}{t}$. For the function $P(t)=88 + 19\ln(t)$, using the sum - rule of differentiation $\frac{d}{dt}(u + v)=\frac{d}{dt}(u)+\frac{d}{dt}(v)$ where $u = 88$ and $v=19\ln(t)$.

Step2: Differentiate each term

$\frac{d}{dt}(88)=0$ and $\frac{d}{dt}(19\ln(t))=19\times\frac{d}{dt}(\ln(t))$. Since $\frac{d}{dt}(\ln(t))=\frac{1}{t}$, then $\frac{d}{dt}(19\ln(t))=\frac{19}{t}$. So $P'(t)=\frac{19}{t}$.

Step3: Evaluate $P'(5)$

Substitute $t = 5$ into $P'(t)$. We get $P'(5)=\frac{19}{5}=3.80$.

Answer:

a) $P'(t)=\frac{19}{t}$
b) $P'(5)=3.80$