Answer
The rate of change of the function $m(p)=4p+p^2$ is equal to $4+2p$ and $\displaystyle\left.\frac{dm}{dp}\right|_{p=-2}=0$
Work Step by Step
\[m(p)=4p+p^2\]
The derivative of the function $m(p)=4p+p^2$ is equal to
\[\frac{dm}{dp}=\lim_{h\rightarrow 0}\frac{m(p+h)-m(p)}{h}\]
\[\Rightarrow \frac{dm}{dp}=\lim_{h\rightarrow 0}\frac{[4(p+h)+(p+h)^2]-[4p+p^2]}{h}\]
\[\Rightarrow \frac{dm}{dp}=\lim_{h\rightarrow 0}\frac{[(4p+4h)+(p^2+h^2+2ph)]-[4p+p^2]}{h}\]
\[\Rightarrow \frac{dm}{dp}=\lim_{h\rightarrow 0}\frac{h^2+4h+2ph}{h}\]
\[\Rightarrow \frac{dm}{dp}=\lim_{h\rightarrow 0}\frac{(h+4+2p)h}{h}\]
\[\Rightarrow \frac{dm}{dp}=\lim_{h\rightarrow 0}(h+4+2p)\]
\[\Rightarrow \frac{dm}{dp}=0+4+2p=4+2p\]
Substitute $p=-2$
\[\left.\frac{dm}{dp}\right|_{p=-2}=4+2(-2)=0\]
Hence , The rate of change of the function $m(p)=4p+p^2$ is equal to $4+2p$ and $\displaystyle\left.\frac{dm}{dp}\right|_{p=-2}=0$