Hello!

Because the task contains expressions `dx / dt` and `dy / dt,` we know that `x` and `y` are functions of `t :`

`x = x ( t ), y = y ( t ) .`

Here `t` is an independent variable.

Also, an equation that connects `x` and `y` is given:

`y = 2 x ^ 2 + 1 .`

This equation is true for any value of the variable `t : `

`y ( t ) = 2 (x( t ) ) ^ 2 + 1 .`

And we are given that `x ( t_0 ) = -2` and `dy / dt (t_0) = -1` for some `t_0 .`

Differentiate the equation with respect to `t :`

`(d / (d t)) ( y ( t ) ) = ( d / (dt) ) (2( x ( t ) ) ^ 2 + 1),`

or

`( (dy) / (dt) ) ( t ) = 4 ( (dx) / (dt) ) ( t ) * x ( t ) ,`

and substitute `t = t_0 : `

`( (dy) / (dt) ) ( t_0 ) = 4 ( (dx) / (dt) ) ( t_0 ) * x ( t_0 ) .`

Which gives `-1 = 4 ( (dx) / (d t) ) ( t_0 ) * ( - 2 )` and finally

`( (dx) / (dt) ) ( t_0 ) = - 1 / ( - 8 ) = 1 / 8 = 0.125 .`

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