Answer:
![(dx)/(dt)=0](https://img.qammunity.org/2021/formulas/mathematics/high-school/dwx2p6gw6yyq0ceux308yf1qrtldkdikav.png)
Explanation:
We have:
![y^2+xy-3x=8](https://img.qammunity.org/2021/formulas/mathematics/high-school/l4l3f653slm46um0r3xvubwgr7ghnh8omt.png)
Where both x and y are functions of t.
To find our solution, let's first take the derivative of both sides with respect to t:
![(d)/(dt)[y^2+xy-3x]=(d)/(dt)[8]](https://img.qammunity.org/2021/formulas/mathematics/high-school/33celdvr0mya1dcgtn0iebut84tvjwi93z.png)
Expand:
![(d)/(dt)[y^2]+(d)/(dt)[xy]+(d)/(dt)[-3x]=(d)/(dt)[8]](https://img.qammunity.org/2021/formulas/mathematics/high-school/9ls0yg1jwpficns2lc768ip1234nyci39a.png)
Differentiate. We must differentiate implicitly. Also, for the second term, we must use the product rule. So:
![(2y(dy)/(dt))+((dx)/(dt)y+x(dy)/(dt))-(3(dx)/(dt))=0](https://img.qammunity.org/2021/formulas/mathematics/high-school/p0hcbefvjedy5r5kwrvygxrpytw7gx0x91.png)
Simplify:
![2y(dy)/(dt)+(dx)/(dt)y+x(dy)/(dt)-3(dx)/(dt)=0](https://img.qammunity.org/2021/formulas/mathematics/high-school/op6lgsdg8m5u3o35ofhhvqhsis3mcn2bd0.png)
We know that dy/dt is 3 when x is -4 and y is 2.
So, to find dx/dt, substitute 3 for dy/dt, -4 for x, and 2 for y. This yields:
![2(2)(3)+(dx)/(dt)(2)+(-4)(3)-3(dx)/(dt)=0](https://img.qammunity.org/2021/formulas/mathematics/high-school/5feq9nuz0rm22d3nd7qbho50iicen569g3.png)
Simplify:
![12+2(dx)/(dt)-12-3(dx)/(dt)=0](https://img.qammunity.org/2021/formulas/mathematics/high-school/2bspgd1cym1r9vec7v199o4e6uehagsnqa.png)
Simplify:
![-(dx)/(dt)=0](https://img.qammunity.org/2021/formulas/mathematics/high-school/1vf4xw4cyp1xabg4gtn46ikhfos7h48t06.png)
Divide both sides by -1:
![(dx)/(dt)=0](https://img.qammunity.org/2021/formulas/mathematics/high-school/dwx2p6gw6yyq0ceux308yf1qrtldkdikav.png)
最终答案是零 :)