Question

If 3x^2y+2x=-32 and dy/dt=-4 when x=2 and y=-3, how do you find dx/dt?

Answer 1

The chain rule implies `dx/dt = (dy/dt)/(dy/dx)" [1]"`
Use implicit differentiation to obtain `dy/dx`
Evaluate `dy/dx` at `(2,-3)`
Substitute the result and `dy/dt` into equation [1]

Explanation:

Given: `3x^2y+2x=-32`

Verify that the point `(2,-3)` lies on the curve:

`3(2)^2(-3)+2(2)=-32`

Use implicit differentiation:

`6xy+3x^2dy/dx + 2 = 0`

`3x^2dy/dx = -2-6xy`

`dy/dx = (-2-6xy)/(3x^2)`

Evaluate `dy/dx` at `(2,-3)`

`dy/dx = (-2-6(2)(-3))/(3(2)^2)`

`dy/dx = 17/6`

Substitute `dy/dx = 17/6` and `dy/dt = -4` into equation [1]:

`dx/dt = (-4)/(17/6)`

`dx/dt = -24/17`