[tex]$\frac{dy}{dt}=-40[/tex]
Solution:
Given data:
[tex]y=-2 x^{2}-5[/tex] and [tex]\frac{dx}{dt}=-5[/tex]
To find [tex]\frac{dy}{dt}[/tex]:
[tex]y=-2 x^{2}-5[/tex]
Differentiate y with respect to t.
[tex]$\frac{dy}{dt}=\frac{d}{dt}(-2x^2-5)[/tex]
[tex]$\frac{dy}{dt}=\frac{d}{dt}(-2x^2)-\frac{d}{dt}(5)[/tex]
Apply the differentiation rule: [tex]\frac{d}{d x}\left(x^{n}\right)=n \cdot x^{n-1}[/tex]
[tex]$\frac{dy}{dt}=2(-2x^{2-1})\cdot \frac{dx}{dt} -\frac{d}{dt}(5)[/tex]
[tex]$\frac{dy}{dt}=-4x\cdot \frac{dx}{dt} -\frac{d}{dt}(5)[/tex]
Apply the differentiation rule: [tex]\frac{d}{dt}a=0[/tex]
[tex]$\frac{dy}{dt}=-4x\cdot \frac{dx}{dt} -0[/tex]
[tex]$\frac{dy}{dt}=-4x\cdot \frac{dx}{dt}[/tex]
At x = –2 and [tex]\frac{dx}{dt}=-5[/tex]
[tex]$\frac{dy}{dt}=-4(-2)\cdot (-5)[/tex]
[tex]$\frac{dy}{dt}=-40[/tex]
Therefore, [tex]\frac{dy}{dt}=-40[/tex].
