Question

Find dy/dt at x = -2 if y = -2x^2 - 5 and dx/dt = -5

dy/dt = ???

Answer 1

`$(dy)/(dt)=-40`

Solution:

Given data:

`y=-2 x^(2)-5` and
`(dx)/(dt)=-5`

To find
`(dy)/(dt)`:

`y=-2 x^(2)-5`

Differentiate y with respect to t.

`$(dy)/(dt)=(d)/(dt)(-2x^2-5)`

`$(dy)/(dt)=(d)/(dt)(-2x^2)-(d)/(dt)(5)`

Apply the differentiation rule:
`(d)/(d x)\left(x^(n)\right)=n \cdot x^(n-1)`

`$(dy)/(dt)=2(-2x^(2-1))\cdot (dx)/(dt) -(d)/(dt)(5)`

`$(dy)/(dt)=-4x\cdot (dx)/(dt) -(d)/(dt)(5)`

Apply the differentiation rule:
`(d)/(dt)a=0`

`$(dy)/(dt)=-4x\cdot (dx)/(dt) -0`

`$(dy)/(dt)=-4x\cdot (dx)/(dt)`

At x = –2 and
`(dx)/(dt)=-5`

`$(dy)/(dt)=-4(-2)\cdot (-5)`

`$(dy)/(dt)=-40`

Therefore,
`(dy)/(dt)=-40`.