Question

If y = 3x^3 - 2x and dx/dt equals 3, find dy/dt when x = -2. Give only the numerical answer. For example, if dy, dt = 4, type only 4.

Answer 1

`(dy)/(dt) = 102`

Explanation:

step(i)

Given function y = 3 x³ - 2 x ... (i)

Differentiating equation (i) with respective to 'x'

`(dy)/(dx) = 3 (3 x^(2) ) - 2(1)`

Given x = -2

`((dy)/(dx) ) x_(=-2) = 3 (3 (-2)^(2) ) - 2(1)`

`((dy)/(dx) ) x_(=-2) = 36 -2 =34`

Step(ii):-

we know that

`(dy)/(dx) = ((dy)/(dt) )/((dx)/(dt) )`

Given
`(dx)/(dt) = 3 and (dy)/(dx) = 34`

`(dy)/(dx) = ((dy)/(dt) )/((dx)/(dt) )`

substitute values
`(dx)/(dt) = 3 and (dy)/(dx) = 34`

⇒
`34 = ((dy)/(dt) )/(3 )`

cross multiplication , we get

`(dy)/(dt) = 34( 3 ) = 102`

Final answer:-

`(dy)/(dt) = 34( 3 ) = 102`