$f^{\prime }(x)=f(x)+x$

$f^{\prime }(x)=\frac{2}{f(x)}.$

1. Suppose $f^{\prime }(t)=3f(t)+5$ and $f(3)=-1$. Find $f^{\prime }(3)$.
   
   $2$

   ans
2. Suppose $f^{\prime }(t)=t+f(t)$, and $f(7)=1$. Find $f^{\prime }(7)$.
   
   $8$

   ans
3. Suppose $f^{\prime }(t)=\frac{1}{\sqrt{f(t)}}$ and $f(0)=9$. Find $f^{\prime }(0)$.
   
   $\frac{1}{3}$

   ans
4. Suppose $f^{\prime }(t)=e^{-f(t)}$ and $f(0)=1$. Find $f^{\prime }(1)$.
   
   Not enough information.

   ans

1. A function is growing at a rate equal to twice the function value.
   
   $f^{\prime }(t)=2f(t)$

   ans
2. A function is growing at a rate equal to the square root of the function value.
   
   $f^{\prime }(t)=\sqrt{f(t)}$

   ans
3. A function is growing at a rate equal to $t$ times the function value.
   
   $f^{\prime }(t)=tf(t)$

   ans
4. A function is accelerating at a rate equal to the sum of the function value and how quickly the function is growing.
   
   $f^{″}(t)=f^{\prime }(t)+f(t)$.

   ans

1. Differential equation $f^{\prime }(t)=f(t)+3$, solution $f(t)=3e^t-3$.
   
   $$\begin{array}{rl}{f}^{\prime }(t)& =f(t)+3\\ \frac{d}{dt}(3e^t–3)& =(3e^t–3)+3\\ 3e^t& =3e^t\end{array}$$

   ans
2. Differential equation $f^{\prime }(t)=4\sqrt{f(t)}$, solution $f(t)=4t^2$.
   
   $$\begin{array}{rl}{f}^{\prime }(t)& =4\sqrt{f(t)}\\ \frac{d}{dt}(4t^2)& =4\sqrt{4t^2}\\ 8t& =4(2t)\\ 8t& =8t.\end{array}$$

   ans
3. Differential equation $f^{\prime }(t)=(f(t){)}^2$, solution $f(t)=-t^{-1}$.
   
   $$\begin{array}{rl}{f}^{\prime }(t)& =(f(t){)}^2\\ \frac{d}{dt}(-t^{-1})& =(-t^{-1}{)}^2\\ t^{-2}& =t^{-2}\end{array}$$

   ans
4. Differential equation $f^{\prime }(t)=e^{-f(t)}$, solution $f(t)=\ln \left(t\right)$.
   
   $$\begin{array}{rl}{f}^{\prime }(t)& =e^{-f(t)}\\ \frac{d}{dt}\ln \left(t\right)& =e^{-\ln \left(t\right)}\\ \frac{1}{t}& =\frac{1}{e^{\ln \left(t\right)}}\\ \frac{1}{t}& =\frac{1}{t}\end{array}$$

   ans